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Exercise 7.2

Integrate the functions in Exercises 1 to 37:

{"font":{"size":12,"family":"Arial","color":"#000000"},"id":"1-0-0","code":"$1.\\,\\frac{2x}{1+x^{2}}$","type":"$","ts":1600065644874,"cs":"AiyagMGEXpS/z/mzSi/0kg==","size":{"width":60,"height":25}}

Solun:- Let f(x) = {"type":"$","id":"1-1","font":{"family":"Arial","color":"#000000","size":12},"code":"$\\frac{2x}{1+x^{2}}$","ts":1600065672411,"cs":"WZUg0FuFXO7mRibrzWHisQ==","size":{"width":40,"height":25}}

Integrate f(x):-

{"type":"$","id":"2-0-0","font":{"family":"Arial","color":"#000000","size":12},"code":"$\\int_{}^{}f\\left(x\\right)dx=\\int_{}^{}\\frac{2x}{1+x^{2}}dx$","ts":1600065718003,"cs":"jAT0PNXcR9FMxX5YMPsP2Q==","size":{"width":188,"height":25}}

Let 1 + x2 = t

Differentiate w.r.t to t:-

⇒ 2x.dx = dt

{"code":"$\\int_{}^{}f\\left(x\\right)dx=\\int_{}^{}\\frac{dt}{t}$","id":"2-1-0-0","font":{"family":"Arial","size":12,"color":"#000000"},"type":"$","ts":1600337939236,"cs":"xix1bxag2lQLMsDdPYk9kQ==","size":{"width":145,"height":24}}

We know that:-

{"code":"$\\int_{}^{}\\frac{1}{x}dx=\\log_{}\\left|x\\right|+c$","id":"3-0","type":"$","font":{"color":"#000000","size":12,"family":"Arial"},"ts":1600065893065,"cs":"FHney6YeAHnbYhVDbCPHIQ==","size":{"width":168,"height":24}}

{"type":"$","id":"2-1-1-0-0","code":"$\\int_{}^{}f\\left(x\\right)dx=\\log_{}\\left|t\\right|+C$","font":{"size":12,"family":"Arial","color":"#000000"},"ts":1600337983744,"cs":"qrhdA7ODiTwtFwo9vm3XXA==","size":{"width":192,"height":22}}

Put the value of t:-

{"type":"$","id":"2-1-1-1-0","font":{"color":"#000000","size":12,"family":"Arial"},"code":"$\\int_{}^{}f\\left(x\\right)dx=\\log_{}\\left|1+x^{2}\\right|+C$","ts":1600066101740,"cs":"aLoS9fTHySb/URcDXCgMpA==","size":{"width":241,"height":24}}

{"id":"1-2-0","code":"$2.\\,\\frac{\\left(\\log_{}x\\right)^{2}}{x}$","font":{"size":12,"family":"Arial","color":"#000000"},"type":"$","ts":1600071878271,"cs":"XMS5tzYcsjx1NCeH4VkAtw==","size":{"width":74,"height":30}}

Solun:- Let f(x) = {"type":"$","font":{"size":10,"color":"#000000","family":"Arial"},"id":"1-3-0","code":"$\\frac{\\left(\\log_{}x\\right)^{2}}{x}$","ts":1600073449273,"cs":"ITeOdloz6Eg7wN31VLxa4w==","size":{"width":42,"height":24}}

Integrate f(x):-

{"font":{"size":12,"color":"#000000","family":"Arial"},"id":"2-0-1-0","code":"$\\int_{}^{}f\\left(x\\right)dx=\\int_{}^{}\\frac{\\left(\\log_{}x\\right)^{2}}{x}dx$","type":"$","ts":1600072017316,"cs":"BcuXMlZm238SfabZDP0IJA==","size":{"width":201,"height":30}}

Let log x = t

Differentiate w.r.t to t:-

⇒ (1/x).dx = dt

{"code":"$\\int_{}^{}f\\left(x\\right)dx=\\int_{}^{}t^{2}.dt$","font":{"size":12,"color":"#000000","family":"Arial"},"type":"$","id":"2-1-0-1-0","ts":1600072781345,"cs":"mG4RYyxPgo0e0kis9mlVbg==","size":{"width":166,"height":24}}

We know that:-

{"id":"3-1-0","code":"$\\int_{}^{}x^{n}.dx=\\frac{x^{n+1}}{n+1}+c$","type":"$","font":{"color":"#000000","size":12,"family":"Arial"},"ts":1600072887456,"cs":"OE5FeFF61sEdUTk8GBxO2Q==","size":{"width":168,"height":28}}

{"font":{"color":"#000000","size":12,"family":"Arial"},"id":"2-1-1-0-1-0","type":"$","code":"$\\int_{}^{}f\\left(x\\right)dx=\\frac{t^{3}}{3}+C$","ts":1600073038892,"cs":"syLdPKaVQf6ALFYa7zED/A==","size":{"width":165,"height":26}}

Put the value of t:-

{"id":"2-1-1-1-1-0","code":"$\\int_{}^{}f\\left(x\\right)dx=\\frac{\\left(\\log_{}\\left|x\\right|\\right)^{3}}{3}+C$","font":{"family":"Arial","color":"#000000","size":12},"type":"$","ts":1600073345670,"cs":"sx6vvy7qqCK/C9M2a0NuaQ==","size":{"width":208,"height":32}}

{"type":"$","id":"1-2-1-0","code":"$3.\\,\\frac{1}{x+x\\log_{}x}$","font":{"size":12,"family":"Arial","color":"#000000"},"ts":1600073591715,"cs":"82JEvx1oDlk4BoEr34m+nw==","size":{"width":88,"height":28}}

Solun:- Let f(x) = {"code":"$\\frac{1}{x+x\\log_{}x}$","font":{"color":"#000000","family":"Arial","size":10},"type":"$","id":"1-3-1-0","ts":1600073726370,"cs":"qtz4y0SnuEk+KKeVSz1dxw==","size":{"width":53,"height":21}}

Integrate f(x):-

{"code":"\\begin{align*}\n{\\int_{}^{}f\\left(x\\right)dx}&={\\int_{}^{}\\frac{1}{x+x\\log_{}x}dx}\\\\\n{\\int_{}^{}f\\left(x\\right)dx}&={\\int_{}^{}\\frac{1}{x\\left(1+\\log_{}x\\right)}dx}\t\n\\end{align*}","id":"2-0-1-1-0","type":"align*","font":{"size":10,"family":"Arial","color":"#000000"},"ts":1600073963335,"cs":"HuCIXZtep4cTz+XGMoo07g==","size":{"width":212,"height":77}}

Let 1 + log x = t

Differentiate w.r.t to t:-

⇒ (1/x).dx = dt

{"font":{"size":12,"color":"#000000","family":"Arial"},"id":"2-1-0-1-1-0","type":"$","code":"$\\int_{}^{}f\\left(x\\right)dx=\\int_{}^{}\\frac{dt}{t}$","ts":1600074373106,"cs":"lS55Q3zljlonY9sAKXb6SQ==","size":{"width":145,"height":24}}

We know that:-

{"code":"$\\int_{}^{}\\frac{1}{x}.dx=\\log_{}\\left|x\\right|+c$","font":{"family":"Arial","color":"#000000","size":12},"id":"3-1-1","type":"$","ts":1600074490846,"cs":"nKoGh9Z58glihLJGFZEZfQ==","size":{"width":176,"height":24}}

{"code":"$\\int_{}^{}f\\left(x\\right)dx=\\log_{}\\left|t\\right|+C$","type":"$","id":"2-1-1-0-1-1-0","font":{"color":"#000000","size":12,"family":"Arial"},"ts":1600074530360,"cs":"aUAUc3eOEEpRQJE44ZSdQQ==","size":{"width":192,"height":22}}

Put the value of t:-

{"code":"$\\int_{}^{}f\\left(x\\right)dx=\\log_{}\\left|1+\\log_{}x\\right|+C$","id":"2-1-1-1-1-1-0","type":"$","font":{"size":12,"color":"#000000","family":"Arial"},"ts":1600074587463,"cs":"oeHDeLddRW7H1rNHUu6B+g==","size":{"width":261,"height":22}}

{"code":"$4.\\,\\sin x.\\sin\\left(\\cos x\\right)$","font":{"family":"Arial","color":"#000000","size":12},"type":"$","id":"1-2-1-1-0","ts":1600074715077,"cs":"ricnUYj5M7WxIDDEM21ObQ==","size":{"width":154,"height":20}}

Solun:- Let f(x) = {"id":"1-3-1-1-0","code":"$\\sin x.\\sin\\left(\\cos x\\right)$","font":{"color":"#000000","family":"Arial","size":10},"type":"$","ts":1600074739648,"cs":"jrZqR2WHmsv8obVSnLgQHA==","size":{"width":102,"height":16}}

Integrate f(x):-

{"id":"2-0-1-1-1-0","font":{"size":10,"family":"Arial","color":"#000000"},"type":"align*","code":"\\begin{align*}\n{\\int_{}^{}f\\left(x\\right)dx}&={\\int_{}^{}\\sin x.\\sin\\left(\\cos x\\right)dx}\t\n\\end{align*}","ts":1600074794534,"cs":"CowLU98YH8SLfl/R6jqmGA==","size":{"width":228,"height":36}}

Let cos x = t

Differentiate w.r.t to t:-

⇒ -sin x.dx = dt

⇒ sin x.dx = - dt

{"type":"$","code":"$\\int_{}^{}f\\left(x\\right)dx=-\\int_{}^{}\\sin t.dt$","id":"2-1-0-1-1-1-0","font":{"size":11,"color":"#000000","family":"Arial"},"ts":1600074999503,"cs":"n6qW4Tw8h3VwABQzvLVtTQ==","size":{"width":182,"height":20}}

We know that:-

{"code":"$\\int_{}^{}\\sin x.dx=-\\cos x+c$","id":"3-1-2-0","type":"$","font":{"size":12,"family":"Arial","color":"#000000"},"ts":1600075318710,"cs":"NSDhN5TuBVoJsnESYJTbEQ==","size":{"width":212,"height":22}}

{"code":"$\\int_{}^{}f\\left(x\\right)dx=\\cos t+C$","type":"$","id":"2-1-1-0-1-1-1-0","font":{"size":12,"color":"#000000","family":"Arial"},"ts":1600075331066,"cs":"94RwK4N8lJdVxhitDgRADg==","size":{"width":184,"height":22}}

Put the value of t:-

{"font":{"family":"Arial","size":12,"color":"#000000"},"id":"2-1-1-1-1-1-1-0","code":"$\\int_{}^{}f\\left(x\\right)dx=\\cos\\left(\\cos x\\right)+C$","type":"$","ts":1600075344448,"cs":"xG0l/oFrrA4sGq/OqgF93g==","size":{"width":232,"height":22}}

{"id":"1-2-1-1-1-0","font":{"color":"#000000","family":"Arial","size":12},"code":"$5.\\,\\sin \\left(ax+b\\right).\\cos \\left(ax+b\\right)$","type":"$","ts":1600075861047,"cs":"BFr0+o4Wl+9s80T9Pss6/w==","size":{"width":228,"height":20}}

Solun:- Let f(x) = sin(ax+b).cos(ax+b)

Integrate f(x):-

{"type":"align*","code":"\\begin{align*}\n{\\int_{}^{}f\\left(x\\right)dx}&={\\int_{}^{}\\sin \\left(ax+b\\right).\\cos \\left(ax+b\\right)dx}\t\n\\end{align*}","id":"2-0-1-1-1-1-0","font":{"size":10,"family":"Arial","color":"#000000"},"ts":1600075439972,"cs":"NKBC6nXvnRs2VKnPXS3Tzg==","size":{"width":284,"height":36}}

Let ax+b = t

Differentiate w.r.t to t:-

⇒ a.dx = dt

⇒ dx = (1/a).dt

{"code":"\\begin{align*}\n{\\int_{}^{}f\\left(x\\right)dx}&={\\frac{1}{a}\\int_{}^{}\\sin t.\\cos t.dt}\\\\\n{\\int_{}^{}f\\left(x\\right)dx}&={\\frac{1}{2a}\\int_{}^{}2\\sin t.\\cos t.dt}\\\\\n{\\int_{}^{}f\\left(x\\right)dx}&={\\frac{1}{2a}\\int_{}^{}\\sin 2t.dt}\t\n\\end{align*}","id":"2-1-0-1-1-1-1-0","type":"align*","font":{"family":"Arial","color":"#000000","size":10},"ts":1600338763135,"cs":"htICFS4FlCRYXi9FFyh6Og==","size":{"width":228,"height":116}}

We know that:-

{"code":"$\\int_{}^{}\\sin x.dx=-\\cos x+c$","type":"$","id":"3-1-2-1-0","font":{"size":12,"color":"#000000","family":"Arial"},"ts":1600075609899,"cs":"vBc4LSUjSrsldO3h3BGNCw==","size":{"width":212,"height":22}}

{"code":"$\\int_{}^{}f\\left(x\\right)dx=\\frac{1}{2a}\\times\\frac{\\left(-\\cos 2t\\right)}{2}+C$","font":{"family":"Arial","size":12,"color":"#000000"},"type":"$","id":"2-1-1-0-1-1-1-1-0","ts":1600338790225,"cs":"ACrpt/EVcojz7IuT59raXw==","size":{"width":262,"height":28}}

Put the value of t:-

{"id":"2-1-1-1-1-1-1-1-0","code":"$\\int_{}^{}f\\left(x\\right)dx=\\frac{-\\cos2\\left(ax+b\\right)}{4a}+C$","font":{"family":"Arial","size":12,"color":"#000000"},"type":"$","ts":1600338811309,"cs":"5OOSyTnjaEqVPCPQU7+Emw==","size":{"width":241,"height":28}}

{"font":{"color":"#000000","family":"Arial","size":12},"code":"$6.\\,{\\sqrt[]{ax+b}}$","type":"$","id":"1-2-1-1-1-1-0","ts":1600075997738,"cs":"2e9e2TPSCjrxAzVG5SBRLg==","size":{"width":92,"height":21}}

Solun:- Let f(x) = {"type":"$","font":{"family":"Arial","size":10,"color":"#000000"},"id":"4-0","code":"${\\sqrt[]{ax+b}}$","ts":1600076030175,"cs":"AePiCOoC7OBBID19PAP7vQ==","size":{"width":56,"height":16}}

Integrate f(x):-

{"font":{"color":"#000000","size":10,"family":"Arial"},"id":"2-0-1-1-1-1-1-0","type":"align*","code":"\\begin{align*}\n{\\int_{}^{}f\\left(x\\right)dx}&={\\int_{}^{}{\\sqrt[]{ax+b}}.dx}\t\n\\end{align*}","ts":1600076077363,"cs":"WgRW79vOP0rUgx5/Q/kJ1w==","size":{"width":192,"height":36}}

Let ax+b = t2

Differentiate w.r.t to t:-

⇒ a.dx = 2t.dt

⇒ dx = (2t/a).dt

{"id":"2-1-0-1-1-1-1-1-0","type":"align*","code":"\\begin{align*}\n{\\int_{}^{}f\\left(x\\right)dx}&={\\frac{1}{a}\\int_{}^{}t\\times2t.dt}\\\\\n{\\int_{}^{}f\\left(x\\right)dx}&={\\frac{1}{a}\\int_{}^{}2t^{2}.dt}\t\n\\end{align*}","font":{"family":"Arial","size":10,"color":"#000000"},"ts":1600077749369,"cs":"yK/biBZmt6sEf8IZSfMuJg==","size":{"width":184,"height":76}}

We know that:-

{"font":{"family":"Arial","color":"#000000","size":12},"code":"$\\int_{}^{}x^{n}.dx=\\frac{x^{n+1}}{n+1}+c$","type":"$","id":"3-1-2-1-1-0","ts":1600077575199,"cs":"aQbb1PHpXdCQALU7LCpH5Q==","size":{"width":168,"height":28}}

{"code":"$\\int_{}^{}f\\left(x\\right)dx=\\frac{2t^{3}}{3a}+C$","type":"$","font":{"size":12,"family":"Arial","color":"#000000"},"id":"2-1-1-0-1-1-1-1-1-0","ts":1600077768489,"cs":"URiDk+Mji38p6vWp685+eg==","size":{"width":173,"height":26}}

Put the value of t:-

{"id":"2-1-1-1-1-1-1-1-1-0","type":"$","font":{"color":"#000000","family":"Arial","size":12},"code":"$\\int_{}^{}f\\left(x\\right)dx=\\frac{2}{3a}\\left(ax+b\\right)^{\\frac{3}{2}}+C$","ts":1600077670407,"cs":"UlNqgU81rjzodbxJqrxJGg==","size":{"width":254,"height":29}}

{"id":"1-2-1-1-1-1-1-0","font":{"family":"Arial","color":"#000000","size":12},"code":"$7.\\,x{\\sqrt[]{x+2}}$","type":"$","ts":1600077865186,"cs":"vXmbwV+atFwFXBwSfSVILw==","size":{"width":94,"height":21}}

Solun:- Let f(x) = {"type":"$","font":{"color":"#000000","family":"Arial","size":10},"id":"4-1-0","code":"$x{\\sqrt[]{x+2}}$","ts":1600077899189,"cs":"IjbzfdYS2fYAZc8Mrnvzxg==","size":{"width":58,"height":16}}

Integrate f(x):-

{"id":"2-0-1-1-1-1-1-1-0","font":{"family":"Arial","size":10,"color":"#000000"},"code":"\\begin{align*}\n{\\int_{}^{}f\\left(x\\right)dx}&={\\int_{}^{}x{\\sqrt[]{x+2}}.dx}\t\n\\end{align*}","type":"align*","ts":1600077937003,"cs":"Y7+EOsu5OAVhYM74GyzOig==","size":{"width":190,"height":36}}

Let x+2 = t2

Differentiate w.r.t to t:-

⇒ dx = 2t.dt

{"id":"2-1-0-1-1-1-1-1-1-0","font":{"color":"#000000","family":"Arial","size":10},"code":"\\begin{align*}\n{\\int_{}^{}f\\left(x\\right)dx}&={\\int_{}^{}\\left(t^{2}-2\\right)\\times t\\times2t.dt}\\\\\n{\\int_{}^{}f\\left(x\\right)dx}&={\\int_{}^{}\\left(2t^{4}-4t^{2}\\right).dt}\t\n\\end{align*}","type":"align*","ts":1600078080746,"cs":"0O/+R2/ckz6su8bTLUO2Yg==","size":{"width":237,"height":76}}

We know that:-

{"font":{"family":"Arial","color":"#000000","size":12},"code":"$\\int_{}^{}x^{n}.dx=\\frac{x^{n+1}}{n+1}+c$","type":"$","id":"3-1-2-1-1-1","ts":1600077575199,"cs":"lfeEN7f0Za5GLRtVUB8wVg==","size":{"width":168,"height":28}}

{"id":"2-1-1-0-1-1-1-1-1-1-0","type":"$","code":"$\\int_{}^{}f\\left(x\\right)dx=\\frac{2t^{5}}{5}-\\frac{4t^{3}}{3}+C$","font":{"size":12,"family":"Arial","color":"#000000"},"ts":1600078160033,"cs":"jVBX7ka4N2fiXtHu6rspmA==","size":{"width":225,"height":26}}

Put the value of t:-

{"code":"$\\int_{}^{}f\\left(x\\right)dx=\\frac{2}{5}\\left(x+2\\right)^{\\frac{5}{2}}-\\frac{4}{3}\\left(x+2\\right)^{\\frac{3}{2}}+C$","font":{"color":"#000000","family":"Arial","size":10},"type":"$","id":"2-1-1-1-1-1-1-1-1-1-0","ts":1600078270302,"cs":"KD/72yKK0BEXmzhkgZfC/Q==","size":{"width":274,"height":22}}

{"code":"$8.\\,x{\\sqrt[]{1+2x^{2}}}$","font":{"family":"Arial","size":12,"color":"#000000"},"type":"$","id":"1-2-1-1-1-1-1-1-0","ts":1600078402609,"cs":"xH7PPj1ZmQ00tWfBF0lcng==","size":{"width":112,"height":21}}

Solun:- Let f(x) = {"id":"4-1-1-0","type":"$","font":{"color":"#000000","size":10,"family":"Arial"},"code":"$x{\\sqrt[]{1+2x^{2}}}$","ts":1600078430008,"cs":"dDw+F/q8ithJ2v/8THw7tQ==","size":{"width":72,"height":16}}

Integrate f(x):-

{"id":"2-0-1-1-1-1-1-1-1-0","font":{"size":10,"color":"#000000","family":"Arial"},"type":"align*","code":"\\begin{align*}\n{\\int_{}^{}f\\left(x\\right)dx}&={\\int_{}^{}x{\\sqrt[]{1+2x^{2}}}.dx}\t\n\\end{align*}","ts":1600078471521,"cs":"WXor4rApo5/F63wAzYv0sw==","size":{"width":208,"height":36}}

Let 1+2x2 = t2

Differentiate w.r.t to t:-

⇒ 4x.dx = 2t.dt

⇒ dx = (t/2x).dt

Put the value of x:-

{"type":"align*","id":"2-1-0-1-1-1-1-1-1-1-0","code":"\\begin{align*}\n{\\int_{}^{}f\\left(x\\right)dx}&={\\int_{}^{}{\\sqrt[]{\\frac{\\left(t^{2}-1\\right)}{2}}}\\times t\\times \\frac{t}{2{\\sqrt[]{\\frac{\\left(t^{2}-1\\right)}{2}}}}.dt}\\\\\n{\\int_{}^{}f\\left(x\\right)dx}&={\\int_{}^{}{\\sqrt[]{\\frac{\\left(t^{2}-1\\right)}{2}}}\\times \\frac{t^{2}}{{\\sqrt[]{2}}{\\sqrt[]{\\left(t^{2}-1\\right)}}}.dt}\\\\\n{\\int_{}^{}f\\left(x\\right)dx}&={\\int_{}^{}\\frac{1}{{\\sqrt[]{2}}}\\times \\frac{t^{2}}{{\\sqrt[]{2}}}.dt}\\\\\n{\\int_{}^{}f\\left(x\\right)dx}&={\\frac{1}{2}\\int_{}^{}t^{2}.dt}\t\n\\end{align*}","font":{"family":"Arial","size":10,"color":"#000000"},"ts":1600079217777,"cs":"97dp8nf23hzW4n54aFwi4Q==","size":{"width":318,"height":193}}

We know that:-

{"font":{"family":"Arial","color":"#000000","size":12},"code":"$\\int_{}^{}x^{n}.dx=\\frac{x^{n+1}}{n+1}+c$","type":"$","id":"3-1-2-1-1-2","ts":1600077575199,"cs":"K6Qo4zZ3i6k41cGxoxtlQg==","size":{"width":168,"height":28}}

{"font":{"size":12,"color":"#000000","family":"Arial"},"type":"$","id":"2-1-1-0-1-1-1-1-1-1-1-0","code":"$\\int_{}^{}f\\left(x\\right)dx=\\frac{1}{2}\\times\\frac{t^{3}}{3}+C$","ts":1600079284623,"cs":"/YpkCSDcAt04GKiI5ND0DQ==","size":{"width":206,"height":26}}

Put the value of t:-

{"type":"align*","font":{"color":"#000000","size":10,"family":"Arial"},"code":"\\begin{align*}\n{\\int_{}^{}f\\left(x\\right)dx}&={\\frac{\\left(\\left(1+2x^{2}\\right)^{\\frac{1}{2}}\\right)^{3}}{6}+C}\\\\\n{\\int_{}^{}f\\left(x\\right)dx}&={\\frac{1}{6}\\left(1+2x^{2}\\right)^{\\frac{3}{2}}+C}\t\n\\end{align*}","id":"2-1-1-1-1-1-1-1-1-1-1-0","ts":1600079471638,"cs":"FgSUzpk4ytX2YsCVSSJ4bw==","size":{"width":225,"height":93}}

{"type":"$","id":"1-2-1-1-1-1-1-1-1-0","font":{"family":"Arial","size":12,"color":"#000000"},"code":"$9.\\,\\left(4x+2\\right){\\sqrt[]{x^{2}+x+1}}$","ts":1600079608350,"cs":"CfW4wWZh7CpWyYjgemTN+A==","size":{"width":200,"height":24}}

Solun:- Let f(x) = {"code":"$\\left(4x+2\\right){\\sqrt[]{x^{2}+x+1}}$","font":{"family":"Arial","size":10,"color":"#000000"},"type":"$","id":"4-1-1-1-0","ts":1600079656678,"cs":"q0fGUajGSqSnsqFgJ09NeQ==","size":{"width":140,"height":18}}

Integrate f(x):-

{"id":"2-0-1-1-1-1-1-1-1-1-0","code":"\\begin{align*}\n{\\int_{}^{}f\\left(x\\right)dx}&={\\int_{}^{}\\left(4x+2\\right){\\sqrt[]{x^{2}+x+1}}.dx}\\\\\n{\\int_{}^{}f\\left(x\\right)dx}&={2\\int_{}^{}\\left(2x+1\\right){\\sqrt[]{x^{2}+x+1}}.dx}\t\n\\end{align*}","type":"align*","font":{"color":"#000000","family":"Arial","size":10},"ts":1600079748532,"cs":"klYtpB4i82BRfFl0ZPkhQA==","size":{"width":284,"height":76}}

Let x2+x+1 = t2

Differentiate w.r.t to t:-

⇒ (2x+1).dx = 2t.dt

⇒ (2x+1).dx = (2t).dt

{"font":{"family":"Arial","size":10,"color":"#000000"},"code":"\\begin{align*}\n{\\int_{}^{}f\\left(x\\right)dx}&={2\\int_{}^{}t\\times2t.dt}\\\\\n{\\int_{}^{}f\\left(x\\right)dx}&={2\\int_{}^{}2t^{2}.dt}\\\\\n{\\int_{}^{}f\\left(x\\right)dx}&={\\int_{}^{}4t^{2}.dt}\t\n\\end{align*}","id":"2-1-0-1-1-1-1-1-1-1-1-0","type":"align*","ts":1600079888246,"cs":"gdtbOWJLidMUUY9KUd/YJQ==","size":{"width":177,"height":116}}

We know that:-

{"font":{"family":"Arial","color":"#000000","size":12},"code":"$\\int_{}^{}x^{n}.dx=\\frac{x^{n+1}}{n+1}+c$","type":"$","id":"3-1-2-1-1-3","ts":1600077575199,"cs":"tgCDQqNLUNbU6xw4MkPjPA==","size":{"width":168,"height":28}}

{"type":"$","code":"$\\int_{}^{}f\\left(x\\right)dx=\\frac{4t^{3}}{3}+C$","font":{"color":"#000000","size":12,"family":"Arial"},"id":"2-1-1-0-1-1-1-1-1-1-1-1-0","ts":1600079913679,"cs":"DPeM6OugIAYnMg2oSCYM1A==","size":{"width":173,"height":26}}

Put the value of t:-

{"font":{"family":"Arial","color":"#000000","size":10},"id":"2-1-1-1-1-1-1-1-1-1-1-1-0","type":"align*","code":"\\begin{align*}\n{\\int_{}^{}f\\left(x\\right)dx}&={\\frac{4\\left(\\left(x^{2}+x+1\\right)^{\\frac{1}{2}}\\right)^{3}}{3}+C}\\\\\n{\\int_{}^{}f\\left(x\\right)dx}&={\\frac{4}{3}\\left(x^{2}+x+1\\right)^{\\frac{3}{2}}+C}\t\n\\end{align*}","ts":1600080009892,"cs":"yCNcPTICNFYM5HyahvBlHQ==","size":{"width":254,"height":93}}

{"code":"$10.\\,\\frac{1}{x-{\\sqrt[]{x}}}$","font":{"family":"Arial","size":12,"color":"#000000"},"id":"1-2-1-1-1-1-1-1-1-1-0","type":"$","ts":1600080086940,"cs":"7TLF4rGgvY3Ur4Dzi6hSkw==","size":{"width":77,"height":29}}

Solun:- Let f(x) = {"font":{"color":"#000000","size":10,"family":"Arial"},"code":"$\\frac{1}{x-{\\sqrt[]{x}}}$","id":"4-1-1-1-1-0","type":"$","ts":1600080113165,"cs":"0XHYdtpa69zf52o+unHzyw==","size":{"width":37,"height":22}}

Integrate f(x):-

{"id":"2-0-1-1-1-1-1-1-1-1-1-0","code":"\\begin{align*}\n{\\int_{}^{}f\\left(x\\right)dx}&={\\int_{}^{}\\frac{1}{x-{\\sqrt[]{x}}}.dx}\\\\\n{\\int_{}^{}f\\left(x\\right)dx}&={\\int_{}^{}\\frac{1}{{\\sqrt[]{x}}\\left({\\sqrt[]{x}}-1\\right)}.dx}\t\n\\end{align*}","type":"align*","font":{"family":"Arial","size":10,"color":"#000000"},"ts":1600080474092,"cs":"Pq6ksb9ymxtCrmJ82vLIuQ==","size":{"width":225,"height":81}}

Let √x - 1 = t

Differentiate w.r.t to t:-

{"code":"\\begin{align*}\n{\\frac{1}{2{\\sqrt[]{x}}}dx}&={dt}\\\\\n{\\frac{1}{{\\sqrt[]{x}}}dx\\,}&={2dt}\t\n\\end{align*}","type":"align*","id":"5-0","font":{"family":"Arial","size":10,"color":"#000000"},"ts":1600081213456,"cs":"8MZn9e25bqVhTDfE/e5s7w==","size":{"width":97,"height":80}}

{"font":{"color":"#000000","size":10,"family":"Arial"},"code":"\\begin{align*}\n{\\int_{}^{}f\\left(x\\right)dx}&={\\int_{}^{}\\frac{2.dt}{t}}\t\n\\end{align*}","id":"2-1-0-1-1-1-1-1-1-1-1-1-0","type":"align*","ts":1600081290988,"cs":"sS9v/6fjQkhN/u3eA2r7JA==","size":{"width":140,"height":36}}

We know that:-

{"code":"$\\int_{}^{}\\frac{1}{x}.dx=\\log_{}\\left|x\\right|+c$","id":"3-1-2-1-1-4-0","font":{"family":"Arial","color":"#000000","size":12},"type":"$","ts":1600081326525,"cs":"q406vnuqxcYfe3vGLY8uvQ==","size":{"width":176,"height":24}}

{"code":"$\\int_{}^{}f\\left(x\\right)dx=2\\log_{}\\left|t\\right|+C$","type":"$","font":{"color":"#000000","size":12,"family":"Arial"},"id":"2-1-1-0-1-1-1-1-1-1-1-1-1-0","ts":1600081497174,"cs":"f2pEIX36zhghRKZ9o04adw==","size":{"width":205,"height":22}}

Put the value of t:-

{"type":"align*","font":{"family":"Arial","color":"#000000","size":10},"id":"2-1-1-1-1-1-1-1-1-1-1-1-1-0","code":"\\begin{align*}\n{\\int_{}^{}f\\left(x\\right)dx}&={2\\log_{}\\left|{\\sqrt[]{x}}-1\\right|+C}\t\n\\end{align*}","ts":1600081473443,"cs":"KPOu9F7SzQ3Jgnl3hy59Tg==","size":{"width":210,"height":36}}

{"type":"$","code":"$11.\\,\\frac{x}{{\\sqrt[]{x+4}}},x>0$","id":"1-2-1-1-1-1-1-1-1-1-1-0","font":{"color":"#000000","family":"Arial","size":12},"ts":1600081589967,"cs":"/USdPr54DPxPlumthD9PnA==","size":{"width":134,"height":25}}

Solun:- Let f(x) = {"code":"$\\frac{x}{{\\sqrt[]{x+4}}}$","id":"4-1-1-1-1-1-0","font":{"size":10,"color":"#000000","family":"Arial"},"type":"$","ts":1600081611204,"cs":"nWWJ52L8GkCrFD1HzSS9tA==","size":{"width":36,"height":20}}

Integrate f(x):-

{"font":{"color":"#000000","family":"Arial","size":10},"type":"align*","id":"2-0-1-1-1-1-1-1-1-1-1-1-0","code":"\\begin{align*}\n{\\int_{}^{}f\\left(x\\right)dx}&={\\int_{}^{}\\frac{x}{{\\sqrt[]{x+4}}}.dx}\t\n\\end{align*}","ts":1600081657457,"cs":"9fiFSIYgBnpSqvH8A/2+QQ==","size":{"width":188,"height":37}}

Let x + 4 = t2

Differentiate w.r.t to t:-

dx = 2t.dt

{"code":"\\begin{align*}\n{\\int_{}^{}f\\left(x\\right)dx}&={\\int_{}^{}\\frac{\\left(t^{2}-4\\right)}{t}.2t.dt}\\\\\n{\\int_{}^{}f\\left(x\\right)dx}&={2\\int_{}^{}\\left(t^{2}-4\\right)dt}\t\n\\end{align*}","id":"2-1-0-1-1-1-1-1-1-1-1-1-1-0","type":"align*","font":{"size":10,"color":"#000000","family":"Arial"},"ts":1600082824238,"cs":"X8Bk/V5V3SWhddXsjsZ4Og==","size":{"width":206,"height":80}}

We know that:-

{"id":"3-1-2-1-1-4-1-0","code":"$\\int_{}^{}x^{n}.dx=\\frac{x^{n+1}}{n+1}+c$","type":"$","font":{"size":12,"color":"#000000","family":"Arial"},"ts":1600082870135,"cs":"TZxov/N0xcby6X7GK5m8mQ==","size":{"width":168,"height":28}}

{"id":"2-1-1-0-1-1-1-1-1-1-1-1-1-1-0","type":"$","font":{"family":"Arial","size":12,"color":"#000000"},"code":"$\\int_{}^{}f\\left(x\\right)dx=2\\left(\\frac{t^{3}}{3}-4t\\right)+C$","ts":1600082926081,"cs":"lgqjGqpuf39ZBeHCPPcDBQ==","size":{"width":242,"height":36}}

Put the value of t:-

{"type":"align*","id":"2-1-1-1-1-1-1-1-1-1-1-1-1-1-0","font":{"color":"#000000","size":10,"family":"Arial"},"code":"\\begin{align*}\n{\\int_{}^{}f\\left(x\\right)dx}&={\\frac{2}{3}\\left(\\left(x+4\\right)^{\\frac{1}{2}}\\right)^{3}-8\\left(x+4\\right)^{\\frac{1}{2}}+C}\\\\\n{\\int_{}^{}f\\left(x\\right)dx}&={\\frac{2}{3}\\left(x+4\\right)^{\\frac{3}{2}}-8\\left(x+4\\right)^{\\frac{1}{2}}+C}\\\\\n{\\int_{}^{}f\\left(x\\right)dx}&={\\left(x+4\\right)^{\\frac{1}{2}}\\left[\\frac{2}{3}\\left(x+4\\right)-8\\right]+C}\\\\\n{\\int_{}^{}f\\left(x\\right)dx}&={\\left(x+4\\right)^{\\frac{1}{2}}\\left[\\frac{2x+8-24}{3}\\right]+C}\\\\\n{\\int_{}^{}f\\left(x\\right)dx}&={\\left(x+4\\right)^{\\frac{1}{2}}\\left[\\frac{2x-16}{3}\\right]+C}\\\\\n{\\int_{}^{}f\\left(x\\right)dx}&={2{\\sqrt[]{\\left(x+4\\right)}}\\left[\\frac{x-8}{3}\\right]+C}\\\\\n{\\int_{}^{}f\\left(x\\right)dx}&={\\frac{2}{3}{\\sqrt[]{\\left(x+4\\right)}}\\times\\left(x-8\\right)+C}\\\\\n\\end{align*}","ts":1600083803220,"cs":"E6VlvyBrfWEuYuYgb439zA==","size":{"width":304,"height":288}}

{"code":"$12.\\,\\left(x^{3}-1\\right)^{\\frac{1}{3}}x^{5}$","id":"1-2-1-1-1-1-1-1-1-1-1-1-0","font":{"color":"#000000","family":"Arial","size":12},"type":"$","ts":1600084090335,"cs":"qjiXrTw5rsT8+V3SK1Kl0A==","size":{"width":134,"height":30}}

Solun:- Let f(x) = {"font":{"family":"Arial","color":"#000000","size":10},"id":"4-1-1-1-1-1-1-0","code":"$\\left(x^{3}-1\\right)^{\\frac{1}{3}}x^{5}$","type":"$","ts":1600084125923,"cs":"VjDuZOgYvvtSL+MUPpE2Mg==","size":{"width":81,"height":24}}

Integrate f(x):-

{"code":"\\begin{align*}\n{\\int_{}^{}f\\left(x\\right)dx}&={\\int_{}^{}\\left(x^{3}-1\\right)^{\\frac{1}{3}}.x^{5}.dx}\\\\\n{\\int_{}^{}f\\left(x\\right)dx}&={\\int_{}^{}\\left(x^{3}-1\\right)^{\\frac{1}{3}}.x^{3}.x^{2}.dx}\t\n\\end{align*}","id":"2-0-1-1-1-1-1-1-1-1-1-1-1","type":"align*","font":{"color":"#000000","size":10,"family":"Arial"},"ts":1600340259249,"cs":"7FsAMsC3Jt5X0zqMuhApaQ==","size":{"width":244,"height":76}}

Let x3 - 1 = t3

Differentiate w.r.t to t:-

⇒ 3x2.dx = 3t2.dt

⇒ x2.dx = (t2).dt

{"code":"\\begin{align*}\n{\\int_{}^{}f\\left(x\\right)dx}&={\\int_{}^{}\\left(t^{3}\\right)^{\\frac{1}{3}}.\\left(t^{3}+1\\right).t^{2}.dt}\\\\\n{\\int_{}^{}f\\left(x\\right)dx}&={\\int_{}^{}t.\\left(t^{3}+1\\right).t^{2}.dt}\\\\\n{\\int_{}^{}f\\left(x\\right)dx}&={\\int_{}^{}t^{3}.\\left(t^{3}+1\\right).dt}\\\\\n{\\int_{}^{}f\\left(x\\right)dx}&={\\int_{}^{}\\left(t^{6}+t^{3}\\right).dt}\t\n\\end{align*}","font":{"family":"Arial","size":10,"color":"#000000"},"type":"align*","id":"2-1-0-1-1-1-1-1-1-1-1-1-1-1-0","ts":1600340210651,"cs":"ysCEVOg6rubK8KW3tC3cWQ==","size":{"width":244,"height":156}}

We know that:-

{"id":"3-1-2-1-1-4-1-1","code":"$\\int_{}^{}x^{n}.dx=\\frac{x^{n+1}}{n+1}+c$","type":"$","font":{"size":12,"color":"#000000","family":"Arial"},"ts":1600082870135,"cs":"cIIAPJVVCf7oCR60cKw+hg==","size":{"width":168,"height":28}}

{"code":"$\\int_{}^{}f\\left(x\\right)dx=\\left(\\frac{t^{7}}{7}+\\frac{t^{4}}{4}\\right)+C$","font":{"family":"Arial","size":12,"color":"#000000"},"id":"2-1-1-0-1-1-1-1-1-1-1-1-1-1-1-0","type":"$","ts":1600086891489,"cs":"PtIOF9LVEW+0jfEofHSH7A==","size":{"width":236,"height":36}}

Put the value of t:-

{"font":{"size":10,"color":"#000000","family":"Arial"},"code":"\\begin{align*}\n{\\int_{}^{}f\\left(x\\right)dx}&={\\left[\\left(\\frac{\\left(x^{3}-1\\right)^{\\frac{1}{3}}}{7}\\right)^{7}+\\left(\\frac{\\left(x^{3}-1\\right)^{\\frac{1}{3}}}{4}\\right)^{4}\\right]+C}\\\\\n{\\int_{}^{}f\\left(x\\right)dx}&={\\left[\\frac{1}{7}\\left(x^{3}-1\\right)^{\\frac{7}{3}}+\\frac{1}{4}\\left(x^{3}-1\\right)^{\\frac{4}{3}}\\right]+C}\t\n\\end{align*}","id":"2-1-1-1-1-1-1-1-1-1-1-1-1-1-1-0","type":"align*","ts":1600086876055,"cs":"0soz7rQ9mAkJ7M2OQf4XZQ==","size":{"width":376,"height":104}}

{"code":"$13.\\,\\frac{x^{2}}{\\left(2+3x^{3}\\right)^{3}}$","font":{"family":"Arial","color":"#000000","size":12},"id":"1-2-1-1-1-1-1-1-1-1-1-1-1-0","type":"$","ts":1600087056758,"cs":"rNmBkSFBcFbcmK2a7jLOEw==","size":{"width":94,"height":33}}

Solun:- Let f(x) = {"id":"4-1-1-1-1-1-1-1-0","font":{"size":10,"family":"Arial","color":"#000000"},"type":"$","code":"$\\frac{x^{2}}{\\left(2+3x^{3}\\right)^{3}}$","ts":1600087077202,"cs":"3VNJAkMRN8oyHTrAVjMDMQ==","size":{"width":50,"height":26}}

Integrate f(x):-

{"code":"\\begin{align*}\n{\\int_{}^{}f\\left(x\\right)dx}&={\\int_{}^{}\\frac{x^{2}}{\\left(2+3x^{3}\\right)^{3}}.dx}\t\n\\end{align*}","type":"align*","font":{"color":"#000000","family":"Arial","size":10},"id":"2-0-1-1-1-1-1-1-1-1-1-1-2-0","ts":1600087160382,"cs":"X1SphzFHOdE8nHESYnblJA==","size":{"width":208,"height":41}}

Let 2+3x3 = t

Differentiate w.r.t to t:-

⇒ 9x2.dx = dt

⇒ x2.dx = (1/9).dt

{"id":"2-1-0-1-1-1-1-1-1-1-1-1-1-1-1-0","code":"\\begin{align*}\n{\\int_{}^{}f\\left(x\\right)dx}&={\\int_{}^{}\\frac{dt}{9\\left(t^{3}\\right)}}\\\\\n{\\int_{}^{}f\\left(x\\right)dx}&={\\frac{1}{9}\\int_{}^{}\\frac{dt}{t^{3}}}\\\\\n{\\int_{}^{}f\\left(x\\right)dx}&={\\frac{1}{9}\\int_{}^{}t^{-3}dt}\t\n\\end{align*}","font":{"size":10,"color":"#000000","family":"Arial"},"type":"align*","ts":1600087577727,"cs":"vhOMfthMgwA5b1j40y0mFg==","size":{"width":160,"height":117}}

We know that:-

{"id":"3-1-2-1-1-4-1-2","code":"$\\int_{}^{}x^{n}.dx=\\frac{x^{n+1}}{n+1}+c$","type":"$","font":{"size":12,"color":"#000000","family":"Arial"},"ts":1600082870135,"cs":"uoyTGlMyraT/nCWmJ7zhfg==","size":{"width":168,"height":28}}

{"font":{"size":10,"color":"#000000","family":"Arial"},"code":"\\begin{align*}\n{\\int_{}^{}f\\left(x\\right)dx}&={\\frac{1}{9}\\left(\\frac{t^{-2}}{-2}\\right)+C}\\\\\n{\\int_{}^{}f\\left(x\\right)dx}&={\\frac{-1}{18}\\left(\\frac{1}{t^{2}}\\right)+C}\t\n\\end{align*}","type":"align*","id":"2-1-1-0-1-1-1-1-1-1-1-1-1-1-1-1-0","ts":1600087705481,"cs":"6TgEUaE1Q2L2SCP3pi+3Bw==","size":{"width":188,"height":81}}

Put the value of t:-

{"type":"align*","id":"2-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-0","code":"\\begin{align*}\n{\\int_{}^{}f\\left(x\\right)dx}&={\\frac{-1}{18}\\left(\\frac{1}{\\left(2+3x^{3}\\right)^{2}}\\right)+C}\t\n\\end{align*}","font":{"size":10,"family":"Arial","color":"#000000"},"ts":1600340622397,"cs":"VAocnmBlU8BAiyD5eOQxSA==","size":{"width":248,"height":46}}

{"id":"1-2-1-1-1-1-1-1-1-1-1-1-1-1-0","type":"$","code":"$14.\\,\\frac{1}{x\\left(\\log_{}x\\right)^{m}},x>0,m\\neq1$","font":{"size":12,"color":"#000000","family":"Arial"},"ts":1600088374904,"cs":"SL1scw0+UVmzu9RX7H3lPw==","size":{"width":220,"height":28}}

Solun:- Let f(x) = {"id":"4-1-1-1-1-1-1-1-1-0","font":{"family":"Arial","size":10,"color":"#000000"},"type":"$","code":"$\\frac{1}{x\\left(\\log_{}x\\right)^{m}}$","ts":1600088465926,"cs":"BHmrQnDPMJ52+/v1v7N97Q==","size":{"width":52,"height":22}}

Integrate f(x):-

{"font":{"size":10,"family":"Arial","color":"#000000"},"type":"align*","id":"2-0-1-1-1-1-1-1-1-1-1-1-2-1-0","code":"\\begin{align*}\n{\\int_{}^{}f\\left(x\\right)dx}&={\\int_{}^{}\\frac{1}{x\\left(\\log_{}x\\right)^{m}}.dx}\t\n\\end{align*}","ts":1600088505726,"cs":"jBXCBPEpz8yeWUtfpDXV7Q==","size":{"width":204,"height":36}}

Let log x = t

Differentiate w.r.t to t:-

⇒ (1/x).dx = dt

{"type":"align*","id":"2-1-0-1-1-1-1-1-1-1-1-1-1-1-1-1-0","font":{"size":10,"color":"#000000","family":"Arial"},"code":"\\begin{align*}\n{\\int_{}^{}f\\left(x\\right)dx}&={\\int_{}^{}\\frac{dt}{t^{m}}}\\\\\n{\\int_{}^{}f\\left(x\\right)dx}&={\\int_{}^{}t^{-m}.dt}\t\n\\end{align*}","ts":1600088667107,"cs":"UOz/s4liFXi9oZjMXg6GLA==","size":{"width":153,"height":76}}

We know that:-

{"id":"3-1-2-1-1-4-1-3","code":"$\\int_{}^{}x^{n}.dx=\\frac{x^{n+1}}{n+1}+c$","type":"$","font":{"size":12,"color":"#000000","family":"Arial"},"ts":1600082870135,"cs":"y4if4KVfb4Rmou2y2qTy7w==","size":{"width":168,"height":28}}

{"code":"\\begin{align*}\n{\\int_{}^{}f\\left(x\\right)dx}&={\\left(\\frac{t^{-m+1}}{-m+1}\\right)+C}\t\n\\end{align*}","id":"2-1-1-0-1-1-1-1-1-1-1-1-1-1-1-1-1-0","font":{"size":10,"color":"#000000","family":"Arial"},"type":"align*","ts":1600088703871,"cs":"9mctfRKeVi4lr0Cm90ghoA==","size":{"width":202,"height":38}}

Put the value of t:-

{"code":"\\begin{align*}\n{\\int_{}^{}f\\left(x\\right)dx}&={\\left(\\frac{\\left(\\log_{}x\\right)^{-m+1}}{-m+1}\\right)+C}\\\\\n{\\int_{}^{}f\\left(x\\right)dx}&={\\left(\\frac{\\left(\\log_{}x\\right)^{1-m}}{1-m}\\right)+C}\t\n\\end{align*}","font":{"color":"#000000","size":10,"family":"Arial"},"id":"2-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-0","type":"align*","ts":1600088848277,"cs":"1tTAh5IvPDjKx7NALqxndg==","size":{"width":229,"height":100}}

{"code":"$15.\\,\\frac{x}{9-4x^{2}}$","font":{"size":12,"color":"#000000","family":"Arial"},"type":"$","id":"1-2-1-1-1-1-1-1-1-1-1-1-1-1-1-0","ts":1600156455971,"cs":"SVt5XbHRhT230p8TybypgA==","size":{"width":77,"height":22}}

Solun:- Let f(x) = {"id":"4-1-1-1-1-1-1-1-1-1-0","code":"$\\frac{x}{9-4x^{2}}$","font":{"color":"#000000","size":10,"family":"Arial"},"type":"$","ts":1600156497973,"cs":"1qy/3hIK1AF8C6Hu/rpm4A==","size":{"width":37,"height":17}}

Integrate f(x):-

{"code":"\\begin{align*}\n{\\int_{}^{}f\\left(x\\right)dx}&={\\int_{}^{}\\frac{x}{9-4x^{2}}.dx}\t\n\\end{align*}","type":"align*","id":"2-0-1-1-1-1-1-1-1-1-1-1-2-1-1-0","font":{"family":"Arial","size":10,"color":"#000000"},"ts":1600156523755,"cs":"pkQXx3tdsmgVPufBuFd2eQ==","size":{"width":188,"height":36}}

Let 9 - 4x2 = t

Differentiate w.r.t to t:-

⇒ -8x.dx = dt

⇒ xdx = (-1/8).dt

{"code":"\\begin{align*}\n{\\int_{}^{}f\\left(x\\right)dx}&={\\int_{}^{}\\frac{-dt}{8t}}\\\\\n{\\int_{}^{}f\\left(x\\right)dx}&={\\frac{-1}{8}\\int_{}^{}\\frac{1}{t}.dt}\t\n\\end{align*}","font":{"color":"#000000","size":10,"family":"Arial"},"id":"2-1-0-1-1-1-1-1-1-1-1-1-1-1-1-1-1-0","type":"align*","ts":1600156804569,"cs":"NFLAS3FLJi/PH6p0NHzNAA==","size":{"width":172,"height":76}}

We know that:-

{"id":"3-1-2-1-1-4-1-4-0","code":"$\\int_{}^{}\\frac{1}{x}.dx=\\log_{}\\left|x\\right|+c$","font":{"color":"#000000","family":"Arial","size":12},"type":"$","ts":1600156993618,"cs":"gzU1tFegM5wa2w9rU+NL6Q==","size":{"width":176,"height":24}}

{"font":{"color":"#000000","size":10,"family":"Arial"},"type":"align*","id":"2-1-1-0-1-1-1-1-1-1-1-1-1-1-1-1-1-1-0","code":"\\begin{align*}\n{\\int_{}^{}f\\left(x\\right)dx}&={\\frac{-1}{8}\\log_{}\\left|t\\right|+C}\t\n\\end{align*}","ts":1600157220340,"cs":"Rq4JxN1vxLBRFk/mK7b/3A==","size":{"width":182,"height":36}}

Put the value of t:-

{"font":{"size":10,"family":"Arial","color":"#000000"},"type":"align*","code":"\\begin{align*}\n{\\int_{}^{}f\\left(x\\right)dx}&={\\frac{-1}{8}\\log_{}\\left|9-4x^{2}\\right|+C}\t\n\\end{align*}","id":"2-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-0","ts":1600157266043,"cs":"uynSvyKenRL0ttwC6156EA==","size":{"width":228,"height":36}}

{"type":"$","code":"$16.\\,e^{2x+3}$","id":"1-2-1-1-1-1-1-1-1-1-1-1-1-1-1-1","font":{"family":"Arial","color":"#000000","size":12},"ts":1600157797923,"cs":"lpmYGO4CLIoS7XF5LNoLhg==","size":{"width":73,"height":16}}

Solun:- Let f(x) = {"font":{"color":"#000000","size":10,"family":"Arial"},"id":"4-1-1-1-1-1-1-1-1-1-1-0","type":"$","code":"$e^{2x+3}$","ts":1600157854001,"cs":"xyEApLxdLou8N6WPmXXU9g==","size":{"width":33,"height":12}}

Integrate f(x):-

{"font":{"size":10,"family":"Arial","color":"#000000"},"code":"\\begin{align*}\n{\\int_{}^{}f\\left(x\\right)dx}&={\\int_{}^{}e^{2x+3}.dx}\t\n\\end{align*}","id":"2-0-1-1-1-1-1-1-1-1-1-1-2-1-1-1-0","type":"align*","ts":1600157906608,"cs":"Lko2wQEmGR6T0TfxGzvV5Q==","size":{"width":165,"height":36}}

Let 2x + 3 = t

Differentiate w.r.t to t:-

⇒ 2.dx = dt

⇒ dx = (1/2).dt

{"code":"\\begin{align*}\n{\\int_{}^{}f\\left(x\\right)dx}&={\\int_{}^{}e^{t}.\\frac{dt}{2}}\\\\\n{\\int_{}^{}f\\left(x\\right)dx}&={\\frac{1}{2}\\int_{}^{}e^{t}.dt}\t\n\\end{align*}","id":"2-1-0-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-0","type":"align*","font":{"size":10,"color":"#000000","family":"Arial"},"ts":1600158277377,"cs":"32OgD01B9yiQ9iLR+HBwsg==","size":{"width":157,"height":76}}

We know that:-

{"id":"3-1-2-1-1-4-1-4-1-0","type":"$","font":{"color":"#000000","size":12,"family":"Arial"},"code":"$\\int_{}^{}e^{x}.dx=e^{x}+c$","ts":1600158308546,"cs":"CUPZEEYTHhXkXzv6ZR0bXQ==","size":{"width":148,"height":22}}

{"type":"align*","font":{"family":"Arial","size":10,"color":"#000000"},"code":"\\begin{align*}\n{\\int_{}^{}f\\left(x\\right)dx}&={\\frac{e^{t}}{2}+C}\t\n\\end{align*}","id":"2-1-1-0-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-0","ts":1600158375725,"cs":"+L/Drb9k9nNUcFCCFbLbCw==","size":{"width":138,"height":37}}

Put the value of t:-

{"type":"align*","code":"\\begin{align*}\n{\\int_{}^{}f\\left(x\\right)dx}&={\\frac{e^{2x+3}}{2}+C}\t\n\\end{align*}","id":"2-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-0","font":{"size":10,"family":"Arial","color":"#000000"},"ts":1600158601429,"cs":"gPuUX2w095HpjVHOx1wXdg==","size":{"width":160,"height":37}}

{"id":"1-2-1-1-1-1-1-1-1-1-1-1-1-1-1-2-0","font":{"color":"#000000","size":12,"family":"Arial"},"type":"$","code":"$17.\\,\\frac{x}{e^{x^{2}}}$","ts":1600158898849,"cs":"ygQpwATISHz3L0Pn1diqyw==","size":{"width":56,"height":24}}

Solun:- Let f(x) = {"id":"4-1-1-1-1-1-1-1-1-1-1-1-0","type":"$","code":"$\\frac{x}{e^{x^{2}}}$","font":{"color":"#000000","family":"Arial","size":10},"ts":1600158919722,"cs":"Zy/PWlUFNLZjAoGYTiNkeg==","size":{"width":21,"height":20}}

Integrate f(x):-

{"id":"2-0-1-1-1-1-1-1-1-1-1-1-2-1-1-1-1-0","code":"\\begin{align*}\n{\\int_{}^{}f\\left(x\\right)dx}&={\\int_{}^{}\\frac{x}{e^{x^{2}}}.dx}\t\n\\end{align*}","type":"align*","font":{"size":10,"color":"#000000","family":"Arial"},"ts":1600158983967,"cs":"xo7zHwOwwHOvxjTEkROiBQ==","size":{"width":157,"height":36}}

Let x2 = t

Differentiate w.r.t to t:-

⇒ 2x.dx = dt

⇒ x.dx = (1/2).dt

{"font":{"size":10,"family":"Arial","color":"#000000"},"code":"\\begin{align*}\n{\\int_{}^{}f\\left(x\\right)dx}&={\\int_{}^{}\\frac{1}{e^{t}}.\\frac{dt}{2}}\\\\\n{\\int_{}^{}f\\left(x\\right)dx}&={\\frac{1}{2}\\int_{}^{}\\frac{1}{e^{t}}.dt}\\\\\n{\\int_{}^{}f\\left(x\\right)dx}&={\\frac{1}{2}\\int_{}^{}e^{-t}.dt}\t\n\\end{align*}","id":"2-1-0-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1","type":"align*","ts":1600159216946,"cs":"dLzqbP18eEo2J5waHmFrbA==","size":{"width":166,"height":116}}

We know that:-

{"type":"$","font":{"color":"#000000","family":"Arial","size":12},"code":"$\\int_{}^{}e^{-x}.dx=-e^{-x}+c$","id":"3-1-2-1-1-4-1-4-1-1-0","ts":1600161553648,"cs":"ZJ6da5aObFfG+JNnCFco1Q==","size":{"width":185,"height":22}}

{"id":"2-1-1-0-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-0","type":"align*","code":"\\begin{align*}\n{\\int_{}^{}f\\left(x\\right)dx}&={\\frac{-e^{-t}}{2}+C}\t\n\\end{align*}","font":{"color":"#000000","family":"Arial","size":10},"ts":1600161573931,"cs":"hZj80YAVAje/SJTsPNrxUA==","size":{"width":160,"height":37}}

Put the value of t:-

{"id":"2-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-0","code":"\\begin{align*}\n{\\int_{}^{}f\\left(x\\right)dx}&={\\frac{-e^{-x^{2}}}{2}+C}\\\\\n{\\int_{}^{}f\\left(x\\right)dx}&={\\frac{-1}{2e^{x^{2}}}+C}\t\n\\end{align*}","font":{"family":"Arial","color":"#000000","size":10},"type":"align*","ts":1600161529405,"cs":"yR86X4ZGkDlltZ1EG6R1cw==","size":{"width":165,"height":80}}

{"font":{"color":"#000000","size":12,"family":"Arial"},"type":"$","id":"1-2-1-1-1-1-1-1-1-1-1-1-1-1-1-2-1-0","code":"$18.\\,\\frac{e^{\\tan^{-1}x}}{1+x^{2}}$","ts":1600161636640,"cs":"ERVx6+27Lt/MRE5feeC4IQ==","size":{"width":81,"height":32}}

Solun:- Let f(x) = {"code":"$\\frac{e^{\\tan^{-1}x}}{1+x^{2}}$","type":"$","font":{"color":"#000000","family":"Arial","size":10},"id":"4-1-1-1-1-1-1-1-1-1-1-1-1-0","ts":1600161680085,"cs":"rPqFNVDh0gwmGcatP1rL3w==","size":{"width":40,"height":24}}

Integrate f(x):-

{"font":{"family":"Arial","color":"#000000","size":10},"code":"\\begin{align*}\n{\\int_{}^{}f\\left(x\\right)dx}&={\\int_{}^{}\\frac{e^{\\tan^{-1}x}}{1+x^{2}}.dx}\t\n\\end{align*}","type":"align*","id":"2-0-1-1-1-1-1-1-1-1-1-1-2-1-1-1-1-1-0","ts":1600161696568,"cs":"piVhwy2AyxIavJGWt64IVA==","size":{"width":181,"height":40}}

Let tan-1x = t

Differentiate w.r.t to t:-

{"type":"$","id":"6-0","code":"$\\left(\\frac{1}{1+x^{2}}\\right)dx=dt$","font":{"family":"Arial","size":11,"color":"#000000"},"ts":1600161738387,"cs":"7oXIu1Ub7Me58OGNEGt6IQ==","size":{"width":117,"height":32}}

{"font":{"family":"Arial","size":10,"color":"#000000"},"id":"2-1-0-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-2-0","code":"\\begin{align*}\n{\\int_{}^{}f\\left(x\\right)dx}&={\\int_{}^{}e^{t}.dt}\t\n\\end{align*}","type":"align*","ts":1600161786355,"cs":"lfB3IkUaLBWWkiauowaD8w==","size":{"width":140,"height":36}}

We know that:-

{"font":{"size":12,"family":"Arial","color":"#000000"},"code":"$\\int_{}^{}e^{x}.dx=e^{x}+c$","type":"$","id":"3-1-2-1-1-4-1-4-1-1-1-0","ts":1600161813378,"cs":"jXBEfUgJymKwoIvy12MEKA==","size":{"width":148,"height":22}}

{"font":{"size":10,"family":"Arial","color":"#000000"},"code":"\\begin{align*}\n{\\int_{}^{}f\\left(x\\right)dx}&={e^{t}+C}\t\n\\end{align*}","type":"align*","id":"2-1-1-0-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-0","ts":1600161849464,"cs":"Pz+p6svott7AYeBju0fdCw==","size":{"width":132,"height":36}}

Put the value of t:-

{"code":"\\begin{align*}\n{\\int_{}^{}f\\left(x\\right)dx}&={e^{\\tan^{-1}x}+C}\t\n\\end{align*}","id":"2-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-0","type":"align*","font":{"color":"#000000","size":10,"family":"Arial"},"ts":1600161884216,"cs":"gzID9nzZ7bI2eFns1GLxJw==","size":{"width":162,"height":36}}

{"type":"$","font":{"color":"#000000","family":"Arial","size":12},"code":"$19.\\,\\frac{e^{2x}-1}{e^{2x}+1}$","id":"1-2-1-1-1-1-1-1-1-1-1-1-1-1-1-2-1-1-0","ts":1600162252189,"cs":"21RkE0p/bZtcGrRllYGzng==","size":{"width":74,"height":28}}

Solun:- Let f(x) = {"font":{"color":"#000000","family":"Arial","size":10},"type":"$","id":"4-1-1-1-1-1-1-1-1-1-1-1-1-1-0","code":"$\\frac{e^{2x}-1}{e^{2x}+1}$","ts":1600162317641,"cs":"G9eqCUZaWSq36xgWv0kvDw==","size":{"width":36,"height":22}}

Integrate f(x):-

{"type":"align*","font":{"size":10,"color":"#000000","family":"Arial"},"id":"2-0-1-1-1-1-1-1-1-1-1-1-2-1-1-1-1-1-1-0","code":"\\begin{align*}\n{\\int_{}^{}f\\left(x\\right)dx}&={\\int_{}^{}\\frac{e^{2x}-1}{e^{2x}+1}.dx}\\\\\n{\\int_{}^{}f\\left(x\\right)dx}&={\\int_{}^{}\\frac{e^{x}\\left(e^{x}-\\frac{1}{e^{x}}\\right)}{e^{x}\\left(e^{x}+\\frac{1}{e^{x}}\\right)}.dx}\\\\\n{\\int_{}^{}f\\left(x\\right)dx}&={\\int_{}^{}\\frac{\\left(e^{x}-e^{-x}\\right)}{\\left(e^{x}+e^{-x}\\right)}.dx}\t\n\\end{align*}","ts":1600163455279,"cs":"vZrE3NgVrenzHy266ugBOQ==","size":{"width":218,"height":132}}

Let ex + e-x = t

Differentiate w.r.t to t:-

⇒ (ex - e-x)dx = dt

{"id":"2-1-0-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-2-1-0","code":"\\begin{align*}\n{\\int_{}^{}f\\left(x\\right)dx}&={\\int_{}^{}\\frac{1}{t}.dt}\t\n\\end{align*}","font":{"color":"#000000","size":10,"family":"Arial"},"type":"align*","ts":1600163936951,"cs":"W7dX0An8sJkrMX0pm1SmpA==","size":{"width":142,"height":36}}

We know that:-

{"code":"$\\int_{}^{}\\frac{1}{x}.dx=\\log_{}\\left|x\\right|+c$","type":"$","font":{"color":"#000000","family":"Arial","size":12},"id":"3-1-2-1-1-4-1-4-1-1-1-1-0","ts":1600164073756,"cs":"bJKWRmKC6nfd/dLtmBEHxQ==","size":{"width":176,"height":24}}

{"type":"align*","code":"\\begin{align*}\n{\\int_{}^{}f\\left(x\\right)dx}&={\\log_{}\\left|t\\right|+C}\t\n\\end{align*}","id":"2-1-1-0-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-0","font":{"color":"#000000","size":10,"family":"Arial"},"ts":1600164101277,"cs":"aqt8yO9fjsSgfbXLPZGhhw==","size":{"width":154,"height":36}}

Put the value of t:-

{"id":"2-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-0","font":{"family":"Arial","size":10,"color":"#000000"},"type":"align*","code":"\\begin{align*}\n{\\int_{}^{}f\\left(x\\right)dx}&={\\log_{}\\left|e^{x}+e^{-x}\\right|+C}\t\n\\end{align*}","ts":1600164311367,"cs":"eVXQ8DynxwG7qOnl7pJ8MA==","size":{"width":208,"height":36}}

{"code":"$20.\\,\\frac{e^{2x}-e^{-2x}}{e^{2x}+e^{-2x}}$","font":{"size":12,"family":"Arial","color":"#000000"},"id":"1-2-1-1-1-1-1-1-1-1-1-1-1-1-1-2-1-1-1-0","type":"$","ts":1600166273116,"cs":"fRYNYW38GsmKPAZHMoytOw==","size":{"width":93,"height":28}}

Solun:- Let f(x) = {"code":"$\\frac{e^{2x}-e^{-2x}}{e^{2x}+e^{-2x}}$","type":"$","id":"4-1-1-1-1-1-1-1-1-1-1-1-1-1-1-0","font":{"family":"Arial","color":"#000000","size":10},"ts":1600166354994,"cs":"SIkOuGWZ08iogY/WKD/qow==","size":{"width":49,"height":22}}

Integrate f(x):-

{"font":{"family":"Arial","color":"#000000","size":10},"type":"align*","id":"2-0-1-1-1-1-1-1-1-1-1-1-2-1-1-1-1-1-1-1-0","code":"\\begin{align*}\n{\\int_{}^{}f\\left(x\\right)dx}&={\\int_{}^{}\\frac{e^{2x}-e^{-2x}}{e^{2x}+e^{-2x}}.dx}\t\n\\end{align*}","ts":1600166427231,"cs":"f2MQkCn6a4UizZAIyexq+A==","size":{"width":206,"height":37}}

Let e2x + e-2x = t

Differentiate w.r.t to t:-

⇒ (2e2x - 2e-2x)dx = dt

⇒ (e2x - e-2x)dx = (1/2).dt

{"id":"2-1-0-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-2-1-1-0","font":{"size":10,"family":"Arial","color":"#000000"},"code":"\\begin{align*}\n{\\int_{}^{}f\\left(x\\right)dx}&={\\int_{}^{}\\frac{1}{t}.\\frac{dt}{2}}\\\\\n{\\int_{}^{}f\\left(x\\right)dx}&={\\frac{1}{2}\\int_{}^{}\\frac{1}{t}.dt}\t\n\\end{align*}","type":"align*","ts":1600166760151,"cs":"sC76lBZBqnBC9m5V/gwEzQ==","size":{"width":160,"height":76}}

We know that:-

{"code":"$\\int_{}^{}\\frac{1}{x}.dx=\\log_{}\\left|x\\right|+c$","type":"$","font":{"color":"#000000","family":"Arial","size":12},"id":"3-1-2-1-1-4-1-4-1-1-1-1-1","ts":1600164073756,"cs":"umsizSFrOJWTL8aSPiKBPg==","size":{"width":176,"height":24}}

{"font":{"size":10,"color":"#000000","family":"Arial"},"code":"\\begin{align*}\n{\\int_{}^{}f\\left(x\\right)dx}&={\\frac{1}{2}\\log_{}\\left|t\\right|+C}\t\n\\end{align*}","id":"2-1-1-0-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-0","type":"align*","ts":1600166888106,"cs":"NQqRQ/s4wB7jwWFYPa/7Ug==","size":{"width":169,"height":36}}

Put the value of t:-

{"type":"align*","code":"\\begin{align*}\n{\\int_{}^{}f\\left(x\\right)dx}&={\\frac{1}{2}\\log_{}\\left|e^{2x}+e^{-2x}\\right|+C}\t\n\\end{align*}","font":{"family":"Arial","size":10,"color":"#000000"},"id":"2-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-0","ts":1600166978906,"cs":"JX6hNxfu2BT35mLl4S4wOA==","size":{"width":233,"height":36}}

{"id":"1-2-1-1-1-1-1-1-1-1-1-1-1-1-1-2-1-1-1-1-0","type":"$","code":"$21.\\,\\tan^{2}\\left(2x-3\\right)$","font":{"color":"#000000","size":12,"family":"Arial"},"ts":1600168540029,"cs":"wxRaVWNTT781haoUZ46N0w==","size":{"width":142,"height":21}}

Solun:- Let f(x) = {"font":{"family":"Arial","size":10,"color":"#000000"},"type":"$","code":"$\\tan^{2}\\left(2x-3\\right)$","id":"4-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-0","ts":1600167271119,"cs":"YQL8QAI7jxsykdenZk9sIg==","size":{"width":85,"height":16}}

Integrate f(x):-

{"font":{"family":"Arial","size":10,"color":"#000000"},"id":"2-0-1-1-1-1-1-1-1-1-1-1-2-1-1-1-1-1-1-1-1-0","code":"\\begin{align*}\n{\\int_{}^{}f\\left(x\\right)dx}&={\\int_{}^{}\\tan^{2}\\left(2x-3\\right).dx}\t\n\\end{align*}","type":"align*","ts":1600167524137,"cs":"C909WNXmKTL6FEGJeFCPBw==","size":{"width":217,"height":36}}

Let 2x - 3 = t

Differentiate w.r.t to t:-

⇒ 2.dx = dt

⇒ dx = (1/2).dt

{"code":"\\begin{align*}\n{\\int_{}^{}f\\left(x\\right)dx}&={\\int_{}^{}\\tan^{2}t.\\frac{dt}{2}}\\\\\n{\\int_{}^{}f\\left(x\\right)dx}&={\\frac{1}{2}\\int_{}^{}\\tan^{2}t.dt}\t\n\\end{align*}","font":{"color":"#000000","family":"Arial","size":10},"id":"2-1-0-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-2-1-1-1-0-0","type":"align*","ts":1600167660131,"cs":"AUQhOvEVAUL6U1fzB5Qv+Q==","size":{"width":184,"height":76}}

We know that:-

⇒ sec2x - tan2x = 1

{"code":"\\begin{align*}\n{\\int_{}^{}f\\left(x\\right)dx}&={\\frac{1}{2}\\left[\\int_{}^{}\\sec^{2}t.dt-\\int_{}^{}1.dx\\right]}\t\n\\end{align*}","id":"2-1-0-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-2-1-1-1-1-0","font":{"family":"Arial","color":"#000000","size":10},"type":"align*","ts":1600168387830,"cs":"/pxMStZAD1h5kToBQPIEBA==","size":{"width":264,"height":37}}

We know that:-

{"id":"3-1-2-1-1-4-1-4-1-1-1-1-2-0","code":"$\\int_{}^{}\\sec^{2}x.dx=\\tan x+c$","type":"$","font":{"size":12,"family":"Arial","color":"#000000"},"ts":1600167999644,"cs":"JRcGDpXTNi6EE1qQvhpEuQ==","size":{"width":204,"height":24}}

{"id":"2-1-0-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-2-1-1-1-2","type":"align*","code":"\\begin{align*}\n{\\int_{}^{}1dx}&={x+C}\t\n\\end{align*}","font":{"color":"#000000","family":"Arial","size":10},"ts":1600168046389,"cs":"amcvbrkOsOkOn8SW2WBdgw==","size":{"width":105,"height":36}}

{"type":"align*","font":{"family":"Arial","color":"#000000","size":10},"code":"\\begin{align*}\n{\\int_{}^{}f\\left(x\\right)dx}&={\\frac{1}{2}\\left(\\tan t-x\\right)+C}\t\n\\end{align*}","id":"2-1-1-0-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-0","ts":1600168436487,"cs":"tMp33fHeb1CLIkSegO2VWA==","size":{"width":205,"height":36}}

Put the value of t:-

{"code":"\\begin{align*}\n{\\int_{}^{}f\\left(x\\right)dx}&={\\frac{1}{2}\\left(\\tan \\left(2x-3\\right)-x\\right)+C}\t\n\\end{align*}","id":"2-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-0","type":"align*","font":{"size":10,"family":"Arial","color":"#000000"},"ts":1600168337148,"cs":"zIxQmH1eB9Gh5rIfpH4HnA==","size":{"width":253,"height":36}}

{"type":"$","font":{"family":"Arial","size":12,"color":"#000000"},"code":"$22.\\,\\sec^{2}\\left(7-4x\\right)$","id":"1-2-1-1-1-1-1-1-1-1-1-1-1-1-1-2-1-1-1-1-1-0","ts":1600168704882,"cs":"ffpyITCdh/16GVL0Qsqewg==","size":{"width":140,"height":21}}

Solun:- Let f(x) = sec2(7 - 4x)

Integrate f(x):-

{"type":"align*","font":{"family":"Arial","color":"#000000","size":10},"code":"\\begin{align*}\n{\\int_{}^{}f\\left(x\\right)dx}&={\\int_{}^{}\\sec^{2}\\left(7-4x\\right).dx}\t\n\\end{align*}","id":"2-0-1-1-1-1-1-1-1-1-1-1-2-1-1-1-1-1-1-1-1-1-0","ts":1600168759082,"cs":"+4VnL+G7pwx0yPG3YouA5A==","size":{"width":214,"height":36}}

Let 7 - 4x = t

Differentiate w.r.t to t:-

⇒ - 4.dx = dt

⇒ dx = (- 1/4).dt

{"font":{"family":"Arial","size":10,"color":"#000000"},"code":"\\begin{align*}\n{\\int_{}^{}f\\left(x\\right)dx}&={\\int_{}^{}\\sec^{2}t.\\frac{-dt}{4}}\\\\\n{\\int_{}^{}f\\left(x\\right)dx}&={\\frac{-1}{4}\\int_{}^{}\\sec^{2}t.dt}\t\n\\end{align*}","id":"2-1-0-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-2-1-1-1-0-1-0","type":"align*","ts":1600168910163,"cs":"RWqdjAE7yXSutJiHzcjeUw==","size":{"width":192,"height":76}}

We know that:-

{"id":"3-1-2-1-1-4-1-4-1-1-1-1-2-1-0","code":"$\\int_{}^{}\\sec^{2}x.dx=\\tan x+c$","type":"$","font":{"family":"Arial","size":11,"color":"#000000"},"ts":1600168960147,"cs":"1xAI/NaY/XX0sVDwoT65zg==","size":{"width":180,"height":20}}

{"id":"2-1-1-0-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-0","code":"\\begin{align*}\n{\\int_{}^{}f\\left(x\\right)dx}&={\\frac{-1}{4}\\tan t+C}\t\n\\end{align*}","type":"align*","font":{"color":"#000000","family":"Arial","size":10},"ts":1600168996528,"cs":"fU1B+0Hvdn3cydg8x0rOeQ==","size":{"width":177,"height":36}}

Put the value of t:-

{"id":"2-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-0","code":"\\begin{align*}\n{\\int_{}^{}f\\left(x\\right)dx}&={\\frac{-1}{4}\\tan \\left(7-4x\\right)+C}\t\n\\end{align*}","type":"align*","font":{"family":"Arial","size":10,"color":"#000000"},"ts":1600169026482,"cs":"fErbLzrbS+vhZWnWPWFDRw==","size":{"width":225,"height":36}}

{"id":"1-2-1-1-1-1-1-1-1-1-1-1-1-1-1-2-1-1-1-1-1-1-0","code":"$23.\\,\\frac{\\sin^{-1}x}{{\\sqrt[]{1-x^{2}}}}$","type":"$","font":{"size":12,"family":"Arial","color":"#000000"},"ts":1600169128378,"cs":"YqU1ymuHLNJSh4AzpzFMhg==","size":{"width":82,"height":32}}

Solun:- Let f(x) = {"font":{"family":"Arial","size":10,"color":"#000000"},"type":"$","code":"$\\frac{\\sin^{-1}x}{{\\sqrt[]{1-x^{2}}}}$","id":"7-0","ts":1600169168033,"cs":"I72p6eAEH5NgVOQJ6fM2+A==","size":{"width":41,"height":24}}

Integrate f(x):-

{"type":"align*","code":"\\begin{align*}\n{\\int_{}^{}f\\left(x\\right)dx}&={\\int_{}^{}\\frac{\\sin^{-1}x}{{\\sqrt[]{1-x^{2}}}}.dx}\t\n\\end{align*}","font":{"color":"#000000","size":10,"family":"Arial"},"id":"2-0-1-1-1-1-1-1-1-1-1-1-2-1-1-1-1-1-1-1-1-1-1-0","ts":1600169184937,"cs":"DD3oy51Iz7O+hl34ExdMxA==","size":{"width":194,"height":40}}

Let sin-1x = t

Differentiate w.r.t to t:-

{"id":"8-0","font":{"color":"#000000","family":"Arial","size":12},"type":"$","code":"$\\frac{1}{{\\sqrt[]{1-x^{2}}}}dx=dt$","ts":1600169234427,"cs":"hEQeera0irX4IlP9oWXHLQ==","size":{"width":120,"height":29}}

{"id":"2-1-0-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-2-1-1-1-0-1-1-0","type":"align*","code":"\\begin{align*}\n{\\int_{}^{}f\\left(x\\right)dx}&={\\int_{}^{}t.dt}\t\n\\end{align*}","font":{"size":10,"color":"#000000","family":"Arial"},"ts":1600169276870,"cs":"i6OIZIuO0moQlLCZ7wnHMg==","size":{"width":133,"height":36}}

We know that:-

{"font":{"size":11,"color":"#000000","family":"Arial"},"type":"$","code":"$\\int_{}^{}x^{n}.dx=\\frac{x^{n+1}}{n+1}+c$","id":"3-1-2-1-1-4-1-4-1-1-1-1-2-1-1-0","ts":1600169321921,"cs":"zUUITddg2M7jpFWxvE5m9w==","size":{"width":149,"height":24}}

{"code":"\\begin{align*}\n{\\int_{}^{}f\\left(x\\right)dx}&={\\frac{t^{2}}{2}+C}\t\n\\end{align*}","type":"align*","font":{"size":10,"color":"#000000","family":"Arial"},"id":"2-1-1-0-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-0","ts":1600169352669,"cs":"EfwPgQ7sJeoPuobB4rXhlQ==","size":{"width":137,"height":37}}

Put the value of t:-

{"id":"2-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-0","type":"align*","font":{"family":"Arial","color":"#000000","size":10},"code":"\\begin{align*}\n{\\int_{}^{}f\\left(x\\right)dx}&={\\frac{\\left(\\sin^{-1}x\\right)^{2}}{2}+C}\t\n\\end{align*}","ts":1600169398980,"cs":"zFX2Pl16afXVrZsqF/mmVw==","size":{"width":192,"height":44}}

{"code":"$24.\\,\\frac{2\\cos x-3\\sin x}{6\\cos x+4\\sin x}$","type":"$","font":{"size":12,"color":"#000000","family":"Arial"},"id":"1-2-1-1-1-1-1-1-1-1-1-1-1-1-1-2-1-1-1-1-1-1-1-0","ts":1600169475135,"cs":"+sq/m8tdJaOpmcSOLw39xA==","size":{"width":126,"height":25}}

Solun:- Let f(x) = {"type":"$","font":{"color":"#000000","family":"Arial","size":10},"id":"7-1-0","code":"$\\frac{2\\cos x-3\\sin x}{6\\cos x+4\\sin x}$","ts":1600169494056,"cs":"vE+qHPbCapHeslfV5BPBog==","size":{"width":76,"height":20}}

Integrate f(x):-

{"code":"\\begin{align*}\n{\\int_{}^{}f\\left(x\\right)dx}&={\\int_{}^{}\\frac{2\\cos x-3\\sin x}{6\\cos x+4\\sin x}.dx}\t\n\\end{align*}","font":{"family":"Arial","color":"#000000","size":10},"id":"2-0-1-1-1-1-1-1-1-1-1-1-2-1-1-1-1-1-1-1-1-1-1-1-0","type":"align*","ts":1600169520244,"cs":"4eYf7+exN6lMViGSbLzPQw==","size":{"width":242,"height":36}}

Let 6cos x + 4sin x = t

Differentiate w.r.t to t:-

⇒ (- 6sin x + 4cos x).dx = dt

⇒ 2(- 3sin x + 2cos x).dx = dt

⇒ (2cos x - 3sin x).dx = (1/2).dt

{"type":"align*","id":"2-1-0-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-2-1-1-1-0-1-1-1-0","font":{"family":"Arial","size":10,"color":"#000000"},"code":"\\begin{align*}\n{\\int_{}^{}f\\left(x\\right)dx}&={\\int_{}^{}\\frac{1}{t}.\\frac{dt}{2}}\t\n\\end{align*}","ts":1600169697860,"cs":"jsesd55LM0bF82Qi+Mf8zA==","size":{"width":149,"height":36}}

{"type":"align*","code":"\\begin{align*}\n{\\int_{}^{}f\\left(x\\right)dx}&={\\frac{1}{2}\\int_{}^{}\\frac{1}{t}.dt}\t\n\\end{align*}","id":"9-0","font":{"size":10,"family":"Arial","color":"#222222"},"ts":1600169734769,"cs":"k0GXrE2wEkjCyjVtAVsqvA==","size":{"width":160,"height":36}}

We know that:-

{"type":"$","font":{"family":"Arial","size":11,"color":"#000000"},"code":"$\\int_{}^{}\\frac{1}{x}.dx=\\log_{}\\left|x\\right|+c$","id":"3-1-2-1-1-4-1-4-1-1-1-1-2-1-1-1-0","ts":1600169804828,"cs":"KT6ge1Q39+5f7592Q88JNg==","size":{"width":157,"height":21}}

{"id":"2-1-1-0-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-0","font":{"size":10,"family":"Arial","color":"#000000"},"type":"align*","code":"\\begin{align*}\n{\\int_{}^{}f\\left(x\\right)dx}&={\\frac{1}{2}\\log_{}\\left|t\\right|+C}\t\n\\end{align*}","ts":1600169850252,"cs":"cOkUa0BNKHhRdefeKHb9ig==","size":{"width":169,"height":36}}

Put the value of t:-

{"id":"2-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-0","font":{"color":"#000000","size":10,"family":"Arial"},"type":"align*","code":"\\begin{align*}\n{\\int_{}^{}f\\left(x\\right)dx}&={\\frac{1}{2}\\log_{}\\left|6\\cos x+4\\sin x\\right|+C}\\\\\n{\\int_{}^{}f\\left(x\\right)dx}&={\\frac{1}{2}\\log_{}\\left|2\\left(3\\cos x+2\\sin x\\right)\\right|+C}\\\\\n{\\int_{}^{}f\\left(x\\right)dx}&={\\frac{1}{2}\\left[\\log_{}\\left|\\left(3\\cos x+2\\sin x\\right)\\right|+\\log_{}\\left|2\\right|\\right]+C}\\\\\n{\\int_{}^{}f\\left(x\\right)dx}&={\\frac{1}{2}\\log_{}\\left|\\left(3\\cos x+2\\sin x\\right)\\right|+c}\t\n\\end{align*}","ts":1600169985446,"cs":"KTDEx7Mt1EfwwY+iyyzBYA==","size":{"width":346,"height":156}}

{"font":{"family":"Arial","size":12,"color":"#000000"},"code":"$25.\\,\\frac{1}{\\cos ^{2}x\\left(1-\\tan x\\right)^{2}}$","type":"$","id":"1-2-1-1-1-1-1-1-1-1-1-1-1-1-1-2-1-1-1-1-1-1-1-1-0","ts":1600170234215,"cs":"F/tvEsvebe5BbYtxaHC8EQ==","size":{"width":140,"height":32}}

Solun:- Let f(x) = {"font":{"color":"#000000","size":10,"family":"Arial"},"type":"$","code":"$\\frac{1}{\\cos ^{2}x\\left(1-\\tan x\\right)^{2}}$","id":"7-1-1-0","ts":1600170210823,"cs":"ytBv+jQXu/CfeHw4AwwV5A==","size":{"width":86,"height":24}}

Integrate f(x):-

{"code":"\\begin{align*}\n{\\int_{}^{}f\\left(x\\right)dx}&={\\int_{}^{}\\frac{1}{\\cos ^{2}x\\left(1-\\tan x\\right)^{2}}.dx}\\\\\n{\\int_{}^{}f\\left(x\\right)dx}&={\\int_{}^{}\\frac{ \\sec^{2}x}{\\left(1-\\tan x\\right)^{2}}.dx}\t\n\\end{align*}","id":"2-0-1-1-1-1-1-1-1-1-1-1-2-1-1-1-1-1-1-1-1-1-1-1-1-0","font":{"color":"#000000","size":10,"family":"Arial"},"type":"align*","ts":1600170155403,"cs":"UGG6JxOfSqCcqnfpSEI84w==","size":{"width":258,"height":86}}

Let 1 - tan x = t

Differentiate w.r.t to t:-

⇒ - sec2x.dx = dt

⇒ sec2x.dx = - dt

{"font":{"size":10,"color":"#000000","family":"Arial"},"code":"\\begin{align*}\n{\\int_{}^{}f\\left(x\\right)dx}&={-\\int_{}^{}\\frac{1}{t^{2}}.dt}\\\\\n{\\int_{}^{}f\\left(x\\right)dx}&={-\\int_{}^{}t^{-2}.dt}\t\n\\end{align*}","id":"2-1-0-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-2-1-1-1-0-1-1-1-1-0","type":"align*","ts":1600170631149,"cs":"GWp4zrh3KL0x8AFKVLN9iA==","size":{"width":164,"height":76}}

We know that:-

{"code":"$\\int_{}^{}x^{n}.dx=\\frac{x^{n+1}}{n+1}+c$","font":{"family":"Arial","size":11,"color":"#000000"},"id":"3-1-2-1-1-4-1-4-1-1-1-1-2-1-1-1-1-0","type":"$","ts":1600170770240,"cs":"XNetN44J14QHkQIEa3mZvw==","size":{"width":149,"height":24}}

{"font":{"family":"Arial","size":10,"color":"#000000"},"type":"align*","id":"2-1-1-0-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-0","code":"\\begin{align*}\n{\\int_{}^{}f\\left(x\\right)dx}&={-\\frac{t^{-1}}{-1}+C}\\\\\n{\\int_{}^{}f\\left(x\\right)dx}&={\\frac{1}{t}+C}\\\\\n\\end{align*}","ts":1600170828098,"cs":"elVDbCWS1EAmPEjkUjgKGg==","size":{"width":158,"height":78}}

Put the value of t:-

{"code":"\\begin{align*}\n{\\int_{}^{}f\\left(x\\right)dx}&={\\frac{1}{1-\\tan x}+C}\t\n\\end{align*}","font":{"color":"#000000","family":"Arial","size":10},"id":"2-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-0","type":"align*","ts":1600170947934,"cs":"9AH0ZZl6yZb0mCxCOmAIwg==","size":{"width":188,"height":36}}

{"font":{"size":12,"family":"Arial","color":"#000000"},"code":"$26.\\,\\frac{\\cos{\\sqrt[]{x}}}{{\\sqrt[]{x}}}$","type":"$","id":"1-2-1-1-1-1-1-1-1-1-1-1-1-1-1-2-1-1-1-1-1-1-1-1-1-0","ts":1600171339566,"cs":"nKggMkFALyneCZ52RV72Pg==","size":{"width":80,"height":33}}

Solun:- Let f(x) = {"id":"7-1-1-1-0","code":"$\\frac{\\cos{\\sqrt[]{x}}}{{\\sqrt[]{x}}}$","type":"$","font":{"family":"Arial","color":"#000000","size":10},"ts":1600171392268,"cs":"4eZ2u+oAU5Ar3b5w5AHBsw==","size":{"width":40,"height":26}}

Integrate f(x):-

{"code":"\\begin{align*}\n{\\int_{}^{}f\\left(x\\right)dx}&={\\int_{}^{}\\frac{\\cos{\\sqrt[]{x}}}{{\\sqrt[]{x}}}.dx}\t\n\\end{align*}","type":"align*","id":"2-0-1-1-1-1-1-1-1-1-1-1-2-1-1-1-1-1-1-1-1-1-1-1-1-1-0","font":{"family":"Arial","size":10,"color":"#000000"},"ts":1600171431285,"cs":"kkp167XIR5tctVz10lmFAQ==","size":{"width":184,"height":40}}

Let √x = t

Differentiate w.r.t to t:-

{"code":"\\begin{align*}\n{\\frac{1}{2{\\sqrt[]{x}}}dx}&={dt}\\\\\n{\\frac{1}{{\\sqrt[]{x}}}dx\\,}&={2.dt}\t\n\\end{align*}","type":"align*","font":{"color":"#222222","size":10,"family":"Arial"},"id":"10-0","ts":1600173092863,"cs":"FnJgHIV3ojc9ZzU1C0otLw==","size":{"width":101,"height":80}}

{"code":"\\begin{align*}\n{\\int_{}^{}f\\left(x\\right)dx}&={\\int_{}^{}2.\\cos t.dt}\\\\\n{\\int_{}^{}f\\left(x\\right)dx}&={2\\int_{}^{}\\cos t.dt}\t\n\\end{align*}","type":"align*","id":"2-1-0-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-2-1-1-1-0-1-1-1-1-1-0","font":{"family":"Arial","size":10,"color":"#000000"},"ts":1600173244973,"cs":"7Tgl8a/FaxUPJFlFdGz0yg==","size":{"width":172,"height":76}}

We know that:-

{"code":"$\\int_{}^{}\\cos x.dx=\\sin x+c$","id":"3-1-2-1-1-4-1-4-1-1-1-1-2-1-1-1-1-1-0","type":"$","font":{"family":"Arial","size":11,"color":"#000000"},"ts":1600173280304,"cs":"OlPAWGTqZ901VaJJd06z6Q==","size":{"width":170,"height":20}}

{"code":"\\begin{align*}\n{\\int_{}^{}f\\left(x\\right)dx}&={2\\sin t+C}\t\n\\end{align*}","type":"align*","font":{"size":10,"color":"#000000","family":"Arial"},"id":"2-1-1-0-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-0","ts":1600173304845,"cs":"NVyDKsBC29nETkQGf39hCw==","size":{"width":157,"height":36}}

Put the value of t:-

{"code":"\\begin{align*}\n{\\int_{}^{}f\\left(x\\right)dx}&={2\\sin {\\sqrt[]{x}}+C}\t\n\\end{align*}","font":{"size":10,"color":"#000000","family":"Arial"},"id":"2-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-0","type":"align*","ts":1600173350641,"cs":"GeLfQtu5bKlnenNOL845TA==","size":{"width":174,"height":36}}

{"type":"$","code":"$27.\\,{\\sqrt[]{\\sin2x.}}\\cos2x$","font":{"size":12,"color":"#000000","family":"Arial"},"id":"1-2-1-1-1-1-1-1-1-1-1-1-1-1-1-2-1-1-1-1-1-1-1-1-1-1-0","ts":1600173575955,"cs":"jDI+oG0cxIwQ90dQZVFwxQ==","size":{"width":158,"height":21}}

Solun:- Let f(x) = {"code":"${\\sqrt[]{\\sin2x.}}\\cos2x$","id":"7-1-1-1-1-0","font":{"color":"#000000","family":"Arial","size":10},"type":"$","ts":1600173636776,"cs":"F8Xm7uOhugQ9hc77Sv05sg==","size":{"width":100,"height":16}}

Integrate f(x):-

{"code":"\\begin{align*}\n{\\int_{}^{}f\\left(x\\right)dx}&={\\int_{}^{}{\\sqrt[]{\\sin2x.}}\\cos2x.dx}\t\n\\end{align*}","type":"align*","font":{"size":10,"family":"Arial","color":"#000000"},"id":"2-0-1-1-1-1-1-1-1-1-1-1-2-1-1-1-1-1-1-1-1-1-1-1-1-1-1-0","ts":1600173674357,"cs":"Qq1y42fAZcscXyPQqHjTCg==","size":{"width":232,"height":36}}

Let sin 2x = t

Differentiate w.r.t to t:-

⇒ 2.cos 2x.dx = dt

⇒ cos 2x.dx = (1/2).dt

{"id":"2-1-0-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-2-1-1-1-0-1-1-1-1-1-1-0","font":{"color":"#000000","size":10,"family":"Arial"},"type":"align*","code":"\\begin{align*}\n{\\int_{}^{}f\\left(x\\right)dx}&={\\frac{1}{2}\\int_{}^{}{\\sqrt[]{t}}.dt}\t\n\\end{align*}","ts":1600174058042,"cs":"I4ptnl8JwxQ9ULdO7n3P0Q==","size":{"width":164,"height":36}}

We know that:-

{"id":"3-1-2-1-1-4-1-4-1-1-1-1-2-1-1-1-1-1-1-0","font":{"color":"#000000","family":"Arial","size":11},"type":"$","code":"$\\int_{}^{}x^{n}.dx=\\frac{x^{n+1}}{n+1}+c$","ts":1600173804723,"cs":"Tfml1m/bBeNMa2JgjGlMNQ==","size":{"width":149,"height":24}}

{"font":{"color":"#000000","family":"Arial","size":10},"code":"\\begin{align*}\n{\\int_{}^{}f\\left(x\\right)dx}&={\\frac{1}{2}\\times\\frac{t^{\\frac{3}{2}}}{\\frac{3}{2}}+C}\\\\\n{\\int_{}^{}f\\left(x\\right)dx}&={\\frac{t^{\\frac{3}{2}}}{3}+C}\t\n\\end{align*}","type":"align*","id":"2-1-1-0-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-0","ts":1600174086480,"cs":"f60cd1/sH6PCCuikwjQf4g==","size":{"width":176,"height":89}}

Put the value of t:-

{"font":{"color":"#000000","family":"Arial","size":10},"type":"align*","id":"2-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-0","code":"\\begin{align*}\n{\\int_{}^{}f\\left(x\\right)dx}&={\\frac{\\left(\\sin2x\\right)^{\\frac{3}{2}}}{3}+C}\t\n\\end{align*}","ts":1600174111756,"cs":"+dmdvYsmOrvP6t8WAFVb8w==","size":{"width":186,"height":42}}

{"font":{"family":"Arial","size":12,"color":"#000000"},"id":"1-2-1-1-1-1-1-1-1-1-1-1-1-1-1-2-1-1-1-1-1-1-1-1-1-1-1-0","code":"$28.\\,\\frac{\\cos x}{{\\sqrt[]{1+\\sin x}}}$","type":"$","ts":1600174564791,"cs":"+k9In/QXVOfBAQdEL8IDrg==","size":{"width":96,"height":25}}

Solun:- Let f(x) = {"font":{"family":"Arial","color":"#000000","size":10},"id":"7-1-1-1-1-1","type":"$","code":"$\\frac{\\cos x}{{\\sqrt[]{1+\\sin x}}}$","ts":1600174175255,"cs":"AQgdM9fQhf2K/zRalOvx2A==","size":{"width":52,"height":20}}

Integrate f(x):-

{"id":"2-0-1-1-1-1-1-1-1-1-1-1-2-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-0","code":"\\begin{align*}\n{\\int_{}^{}f\\left(x\\right)dx}&={\\int_{}^{}\\frac{\\cos x}{{\\sqrt[]{1+\\sin x}}}.dx}\t\n\\end{align*}","font":{"family":"Arial","color":"#000000","size":10},"type":"align*","ts":1600174254251,"cs":"z3J5CP+VWJznvKe467wFag==","size":{"width":210,"height":37}}

Let 1 + sin x = t2

Differentiate w.r.t to t:-

⇒ cos x.dx = 2t.dt

{"font":{"color":"#000000","family":"Arial","size":10},"code":"\\begin{align*}\n{\\int_{}^{}f\\left(x\\right)dx}&={\\int_{}^{}\\frac{2t.dt}{t}}\\\\\n{\\int_{}^{}f\\left(x\\right)dx}&={2\\int_{}^{}1.dt}\t\n\\end{align*}","id":"2-1-0-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-2-1-1-1-0-1-1-1-1-1-1-1-0","type":"align*","ts":1600174334619,"cs":"0GZKXlaJGjOdkU9NtN/I/w==","size":{"width":148,"height":76}}

We know that:-

{"type":"$","code":"$\\int_{}^{}1.dx=x+c$","font":{"family":"Arial","size":11,"color":"#000000"},"id":"3-1-2-1-1-4-1-4-1-1-1-1-2-1-1-1-1-1-1-1-0","ts":1600174387546,"cs":"eRA0yU1ileBApsd+ryvupw==","size":{"width":113,"height":20}}

{"font":{"color":"#000000","family":"Arial","size":10},"type":"align*","id":"2-1-1-0-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-0","code":"\\begin{align*}\n{\\int_{}^{}f\\left(x\\right)dx}&={2t+C}\t\n\\end{align*}","ts":1600174425597,"cs":"P+ePyBnPzRBFsT/N6Gx1AQ==","size":{"width":132,"height":36}}

Put the value of t:-

{"type":"align*","font":{"family":"Arial","size":10,"color":"#000000"},"id":"2-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-0","code":"\\begin{align*}\n{\\int_{}^{}f\\left(x\\right)dx}&={2{\\sqrt[]{1+\\sin x}}+C}\t\n\\end{align*}","ts":1600174458063,"cs":"wLmZ+7gilKgas8UO5OBP+Q==","size":{"width":198,"height":36}}

{"type":"$","code":"$29.\\,\\cot x\\log_{}\\left(\\sin x\\right)$","id":"1-2-1-1-1-1-1-1-1-1-1-1-1-1-1-2-1-1-1-1-1-1-1-1-1-1-1-1-0","font":{"size":12,"family":"Arial","color":"#000000"},"ts":1600174540791,"cs":"8NYj0yLPTX6Vtr5BQSp3qg==","size":{"width":160,"height":20}}

Solun:- Let f(x) = cot x.log(sin x)

Integrate f(x):-

{"id":"2-0-1-1-1-1-1-1-1-1-1-1-2-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-0","type":"align*","font":{"size":10,"family":"Arial","color":"#000000"},"code":"\\begin{align*}\n{\\int_{}^{}f\\left(x\\right)dx}&={\\int_{}^{}\\cot x.\\log_{}\\left(\\sin x\\right).dx}\t\n\\end{align*}","ts":1600174819039,"cs":"FECYYbScg9V4q8AraDY/vw==","size":{"width":236,"height":36}}

Let log (sin x) = t

Differentiate w.r.t to t:-

{"id":"14","code":"$\\frac{1}{\\sin x}\\times \\cos x.dx=dt$","font":{"color":"#000000","size":10,"family":"Arial"},"type":"$","ts":1600344072588,"cs":"Z2rsCRF9x4LTFV9oVq7blQ==","size":{"width":140,"height":18}}

{"code":"$\\cot x.dx=dt$","type":"$","font":{"size":10,"family":"Arial","color":"#000000"},"id":"15","ts":1600344248322,"cs":"Qh40bCIhqNs4auIYylHvMQ==","size":{"width":92,"height":12}}

{"type":"align*","font":{"family":"Arial","size":10,"color":"#000000"},"id":"2-1-0-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-2-1-1-1-0-1-1-1-1-1-1-1-1","code":"\\begin{align*}\n{\\int_{}^{}f\\left(x\\right)dx}&={\\int_{}^{}t.dt}\t\n\\end{align*}","ts":1600344281029,"cs":"dOj1HpMV292kD5Ddwi7YkQ==","size":{"width":133,"height":36}}

We know that:-

{"id":"3-1-2-1-1-4-1-4-1-1-1-1-2-1-1-1-1-1-1-1-1","type":"$","code":"$\\int_{}^{}x^{n}.dx=\\frac{x^{n+1}}{n+1}+c$","font":{"color":"#000000","family":"Arial","size":11},"ts":1600344327170,"cs":"n3uSQElgDB9ib9PDGrrPjg==","size":{"width":149,"height":24}}

{"type":"align*","font":{"family":"Arial","size":10,"color":"#000000"},"code":"\\begin{align*}\n{\\int_{}^{}f\\left(x\\right)dx}&={\\frac{t^{2}}{2}+C}\t\n\\end{align*}","id":"2-1-1-0-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1","ts":1600344356701,"cs":"4mPY/Q2Vg3F3gyQpbTtC5Q==","size":{"width":137,"height":37}}

Put the value of t:-

{"font":{"size":10,"color":"#000000","family":"Arial"},"type":"align*","code":"\\begin{align*}\n{\\int_{}^{}f\\left(x\\right)dx}&={\\frac{\\left[\\log_{}\\left(\\sin x\\right)\\right]^{2}}{2}+C}\t\n\\end{align*}","id":"2-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1","ts":1600344396816,"cs":"43KV5lFWnn15jRSndmwxew==","size":{"width":205,"height":40}}

{"type":"$","code":"$30.\\,\\frac{\\sin x}{1+\\cos x}$","id":"1-2-1-1-1-1-1-1-1-1-1-1-1-1-1-2-1-1-1-1-1-1-1-1-1-1-1-1-1-0","font":{"family":"Arial","size":12,"color":"#000000"},"ts":1600245607008,"cs":"NUa4sYyBdKzISpxRZZBaiQ==","size":{"width":86,"height":25}}

Solun:- Let f(x) = {"code":"$\\frac{\\sin x}{1+\\cos x}$","font":{"family":"Arial","color":"#000000","size":10},"type":"$","id":"11-0","ts":1600246583654,"cs":"BjNzVIgoAlsWWpUlQtjgdA==","size":{"width":44,"height":20}}

Integrate f(x):-

{"id":"2-0-1-1-1-1-1-1-1-1-1-1-2-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-0","font":{"size":10,"color":"#000000","family":"Arial"},"code":"\\begin{align*}\n{\\int_{}^{}f\\left(x\\right)dx}&={\\int_{}^{}\\frac{\\sin x}{1+\\cos x}.dx}\t\n\\end{align*}","type":"align*","ts":1600246605513,"cs":"M9yCra6hRtQPCULlNGpfCA==","size":{"width":198,"height":36}}

Let 1 + cos x = t

Differentiate w.r.t to t:-

⇒ -sin x.dx = dt

⇒ sin x.dx = - dt

{"id":"2-1-0-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-2-1-1-1-0-1-1-1-1-1-1-1-2-0","code":"\\begin{align*}\n{\\int_{}^{}f\\left(x\\right)dx}&={\\int_{}^{}\\frac{-dt}{t}}\t\n\\end{align*}","type":"align*","font":{"color":"#000000","family":"Arial","size":10},"ts":1600246755072,"cs":"W0lMSLiBnlt8k3nPa9PAog==","size":{"width":140,"height":36}}

We know that:-

{"type":"$","id":"3-1-2-1-1-4-1-4-1-1-1-1-2-1-1-1-1-1-1-1-2-0","font":{"color":"#000000","family":"Arial","size":11},"code":"$\\int_{}^{}\\frac{1}{x}.dx=\\log_{}\\left|x\\right|+c$","ts":1600246844099,"cs":"0ldI8Qs4+0u6mIDziCFYjw==","size":{"width":157,"height":21}}

{"id":"2-1-1-0-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-2-0","type":"align*","code":"\\begin{align*}\n{\\int_{}^{}f\\left(x\\right)dx}&={-\\log_{}\\left|t\\right|+C}\t\n\\end{align*}","font":{"color":"#000000","size":10,"family":"Arial"},"ts":1600246869360,"cs":"2mgX7EKuXz2I2vQtYE58rg==","size":{"width":169,"height":36}}

Put the value of t:-

{"font":{"size":10,"family":"Arial","color":"#000000"},"code":"\\begin{align*}\n{\\int_{}^{}f\\left(x\\right)dx}&={-\\log_{}\\left|1+\\cos x\\right|+C}\t\n\\end{align*}","type":"align*","id":"2-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-2-0","ts":1600247152893,"cs":"8sB/T4uMLf6cboTDrWDtPA==","size":{"width":224,"height":36}}

{"code":"$31.\\,\\frac{\\sin x}{\\left(1+\\cos x\\right)^{2}}$","id":"1-2-1-1-1-1-1-1-1-1-1-1-1-1-1-2-1-1-1-1-1-1-1-1-1-1-1-1-1-1-0","type":"$","font":{"size":12,"family":"Arial","color":"#000000"},"ts":1600247257486,"cs":"nPJhgmJy0oHgtokT5K+FbA==","size":{"width":104,"height":32}}

Solun:- Let f(x) = {"type":"$","font":{"family":"Arial","color":"#000000","size":10},"id":"11-1-0","code":"$\\frac{\\sin x}{\\left(1+\\cos x\\right)^{2}}$","ts":1600247286765,"cs":"UsC+URm+u0i3mpvItRUatg==","size":{"width":57,"height":24}}

Integrate f(x):-

{"font":{"size":10,"family":"Arial","color":"#000000"},"code":"\\begin{align*}\n{\\int_{}^{}f\\left(x\\right)dx}&={\\int_{}^{}\\frac{\\sin x}{\\left(1+\\cos x\\right)^{2}}.dx}\t\n\\end{align*}","type":"align*","id":"2-0-1-1-1-1-1-1-1-1-1-1-2-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-0","ts":1600247849056,"cs":"E1h4Kb/JjhsrDOuqWwYd8g==","size":{"width":217,"height":40}}

Let 1 + cos x = t

Differentiate w.r.t to t:-

⇒ -sin x.dx = dt

⇒ sin x.dx = - dt

{"type":"align*","font":{"size":10,"family":"Arial","color":"#000000"},"id":"2-1-0-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-2-1-1-1-0-1-1-1-1-1-1-1-2-1-0","code":"\\begin{align*}\n{\\int_{}^{}f\\left(x\\right)dx}&={\\int_{}^{}\\frac{-dt}{t^{2}}}\t\n\\end{align*}","ts":1600247925073,"cs":"uhmJpsjFrIdaUKRAPTSmog==","size":{"width":140,"height":36}}

We know that:-

{"id":"3-1-2-1-1-4-1-4-1-1-1-1-2-1-1-1-1-1-1-1-2-1-0","font":{"family":"Arial","size":11,"color":"#000000"},"code":"$\\int_{}^{}x^{n}.dx=\\frac{x^{n+1}}{n+1}+c$","type":"$","ts":1600247980579,"cs":"Ztrc7lzTh+LV8kO2gJ8RjQ==","size":{"width":149,"height":24}}

{"font":{"family":"Arial","size":10,"color":"#000000"},"type":"align*","id":"2-1-1-0-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-2-1-0","code":"\\begin{align*}\n{\\int_{}^{}f\\left(x\\right)dx}&={\\frac{-t^{-1}}{-1}+C}\\\\\n{\\int_{}^{}f\\left(x\\right)dx}&={\\frac{1}{t}+C}\t\n\\end{align*}","ts":1600248134183,"cs":"1PvG9KBH+agNI3yCWcOVyw==","size":{"width":158,"height":78}}

Put the value of t:-

{"font":{"family":"Arial","color":"#000000","size":10},"id":"2-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-2-1-0","code":"\\begin{align*}\n{\\int_{}^{}f\\left(x\\right)dx}&={\\frac{1}{1+\\cos x}+C}\t\n\\end{align*}","type":"align*","ts":1600248455362,"cs":"1TNxElqXBHugPDUrJmtPiA==","size":{"width":186,"height":36}}

{"type":"$","font":{"size":14.5,"color":"#000000","family":"Arial"},"id":"1-2-1-1-1-1-1-1-1-1-1-1-1-1-1-2-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-0","code":"$32.\\,\\frac{1}{1+\\cot x}$","ts":1600249162563,"cs":"FxPCdUVcUi00uu/+9fV4Jw==","size":{"width":105,"height":32}}

Solun:- Let f(x) = {"font":{"family":"Arial","color":"#000000","size":11.666666666666668},"type":"$","code":"$\\frac{1}{1+\\cot x}$","id":"11-1-1-1-0","ts":1600249232542,"cs":"w70jSbJPRv8rG2kGW4UKZw==","size":{"width":56,"height":25}}

Integrate f(x):-

{"type":"align*","code":"\\begin{align*}\n{\\int_{}^{}f\\left(x\\right)dx}&={\\int_{}^{}\\frac{1}{1+\\cot x}.dx}\\\\\n{\\int_{}^{}f\\left(x\\right)dx}&={\\int_{}^{}\\frac{1}{1+\\frac{\\cos x}{\\sin x}}.dx}\\\\\n{\\int_{}^{}f\\left(x\\right)dx}&={\\int_{}^{}\\frac{\\sin x}{\\sin x+\\cos x}.dx}\\\\\n{\\int_{}^{}f\\left(x\\right)dx}&={\\frac{1}{2}\\int_{}^{}\\frac{2\\sin x}{\\sin x+\\cos x}.dx}\\\\\n{\\int_{}^{}f\\left(x\\right)dx}&={\\frac{1}{2}\\int_{}^{}\\frac{\\sin x+\\sin x-\\cos x+\\cos x}{\\sin x+\\cos x}.dx}\\\\\n{\\int_{}^{}f\\left(x\\right)dx}&={\\frac{1}{2}\\int_{}^{}\\frac{\\left(\\sin x-\\cos x\\right)+\\left(\\sin x+\\cos x\\right)}{\\sin x+\\cos x}.dx}\\\\\n{\\int_{}^{}f\\left(x\\right)dx}&={\\frac{1}{2}\\left[\\int_{}^{}\\frac{\\left(\\sin x-\\cos x\\right)}{\\sin x+\\cos x}.dx+\\int_{}^{}\\frac{\\left(\\sin x+\\cos x\\right)}{\\sin x+\\cos x}.dx\\right]}\\\\\n\\end{align*}","font":{"family":"Arial","color":"#000000","size":10},"id":"2-0-1-1-1-1-1-1-1-1-1-1-2-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-0","ts":1600254063607,"cs":"8vECMVxN223TpzsXurVhLg==","size":{"width":432,"height":284}}

Let sin x + cos x = t

Differentiate w.r.t to t:-

⇒ (cos x - sin x).dx = dt

⇒ (sin x - cos x).dx = - dt

{"code":"\\begin{align*}\n{\\int_{}^{}f\\left(x\\right)dx}&={\\frac{1}{2}\\left[\\int_{}^{}\\frac{-dt}{t}+\\int_{}^{}1.dx\\right]}\t\n\\end{align*}","type":"align*","id":"2-1-0-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-2-1-1-1-0-1-1-1-1-1-1-1-2-1-1-0","font":{"size":10,"color":"#000000","family":"Arial"},"ts":1600254490210,"cs":"pfQV+83ST6DICHRwHHffLA==","size":{"width":240,"height":37}}

We know that:-

{"id":"3-1-2-1-1-4-1-4-1-1-1-1-2-1-1-1-1-1-1-1-2-1-1-0-0","type":"$","font":{"size":11,"family":"Arial","color":"#000000"},"code":"$\\int_{}^{}\\frac{1}{x}.dx=\\log_{}\\left|x\\right|+c$","ts":1600255066283,"cs":"yPMijasUWweuMj6fO7h9OQ==","size":{"width":157,"height":21}}

{"type":"$","id":"3-1-2-1-1-4-1-4-1-1-1-1-2-1-1-1-1-1-1-1-2-1-1-1-0","font":{"family":"Arial","size":11,"color":"#000000"},"code":"$\\int_{}^{}1.dx=x+c$","ts":1600255089431,"cs":"37ZN2LDBTS6H4Wg/yWtfdQ==","size":{"width":113,"height":20}}

{"code":"\\begin{align*}\n{\\int_{}^{}f\\left(x\\right)dx}&={\\frac{1}{2}\\left[-\\log_{}\\left|t\\right|+x\\right]+C}\t\n\\end{align*}","id":"2-1-1-0-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-2-1-1-0","font":{"size":10,"family":"Arial","color":"#000000"},"type":"align*","ts":1600255149223,"cs":"ocj0Fgq8lZsq2uaPfaVeog==","size":{"width":221,"height":36}}

Put the value of t:-

{"id":"2-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-2-1-1-0","code":"\\begin{align*}\n{\\int_{}^{}f\\left(x\\right)dx}&={\\frac{1}{2}\\left(-\\log_{}\\left|\\sin x+\\cos x\\right|+x\\right)+C}\\\\\n{\\int_{}^{}f\\left(x\\right)dx}&={\\frac{x}{2}-\\frac{1}{2}\\log_{}\\left|\\sin x+\\cos x\\right|+C}\t\n\\end{align*}","font":{"size":10,"family":"Arial","color":"#000000"},"type":"align*","ts":1600255347697,"cs":"xg8hANAwH6WtC6cFcAgM3g==","size":{"width":302,"height":76}}

{"font":{"color":"#000000","size":12,"family":"Arial"},"code":"$33.\\,\\frac{1}{1-\\tan x}$","id":"1-2-1-1-1-1-1-1-1-1-1-1-1-1-1-2-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-0","type":"$","ts":1600255712937,"cs":"MaeDChRtnXDrUUfBu7qevg==","size":{"width":88,"height":25}}

Solun:- Let f(x) = {"font":{"size":11.666666666666668,"family":"Arial","color":"#000000"},"id":"11-1-1-1-1-0","code":"$\\frac{1}{1-\\tan x}$","type":"$","ts":1600255948970,"cs":"cPzBj0rqzO5UIoDFxkBkZA==","size":{"width":58,"height":25}}

Integrate f(x):-

{"font":{"color":"#000000","family":"Arial","size":10},"type":"align*","code":"\\begin{align*}\n{\\int_{}^{}f\\left(x\\right)dx}&={\\int_{}^{}\\frac{1}{1-\\tan x}.dx}\\\\\n{\\int_{}^{}f\\left(x\\right)dx}&={\\int_{}^{}\\frac{1}{1-\\frac{\\sin x}{\\cos x}}.dx}\\\\\n{\\int_{}^{}f\\left(x\\right)dx}&={\\int_{}^{}\\frac{\\cos x}{\\cos x-\\sin x}.dx}\\\\\n{\\int_{}^{}f\\left(x\\right)dx}&={\\frac{1}{2}\\int_{}^{}\\frac{2\\cos x}{\\cos x-\\sin x}.dx}\\\\\n{\\int_{}^{}f\\left(x\\right)dx}&={\\frac{1}{2}\\int_{}^{}\\frac{\\cos x+\\cos x-\\sin x+\\sin x}{\\cos x-\\sin x}.dx}\\\\\n{\\int_{}^{}f\\left(x\\right)dx}&={\\frac{1}{2}\\int_{}^{}\\frac{\\left(\\sin x+\\cos x\\right)+\\left(\\cos x-\\sin x\\right)}{\\cos x-\\sin x}.dx}\\\\\n{\\int_{}^{}f\\left(x\\right)dx}&={\\frac{1}{2}\\left[\\int_{}^{}\\frac{\\left(\\sin x+\\cos x\\right)}{\\cos x-\\sin x}.dx+\\int_{}^{}\\frac{\\left(\\cos x-\\sin x\\right)}{\\cos x-\\sin x}.dx\\right]}\t\n\\end{align*}","id":"2-0-1-1-1-1-1-1-1-1-1-1-2-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-0","ts":1600256688872,"cs":"eMIZ8vmf67VtomyqsyGYYQ==","size":{"width":432,"height":286}}

Let cos x - sin x = t

Differentiate w.r.t to t:-

⇒ (- sin x - cos x).dx = dt

⇒ (sin x + cos x).dx = - dt

{"code":"\\begin{align*}\n{\\int_{}^{}f\\left(x\\right)dx}&={\\frac{1}{2}\\left[\\int_{}^{}\\frac{-dt}{t}+\\int_{}^{}1.dx\\right]}\t\n\\end{align*}","type":"align*","id":"2-1-0-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-2-1-1-1-0-1-1-1-1-1-1-1-2-1-1-1-0","font":{"size":10,"color":"#000000","family":"Arial"},"ts":1600254490210,"cs":"w06zY0jdD3nAuQFXzQy1Dw==","size":{"width":240,"height":37}}

We know that:-

{"id":"3-1-2-1-1-4-1-4-1-1-1-1-2-1-1-1-1-1-1-1-2-1-1-0-1","type":"$","font":{"size":11,"family":"Arial","color":"#000000"},"code":"$\\int_{}^{}\\frac{1}{x}.dx=\\log_{}\\left|x\\right|+c$","ts":1600255066283,"cs":"C56uxPY0XSSX0u++ckXtNw==","size":{"width":157,"height":21}}

{"type":"$","id":"3-1-2-1-1-4-1-4-1-1-1-1-2-1-1-1-1-1-1-1-2-1-1-1-1","font":{"family":"Arial","size":11,"color":"#000000"},"code":"$\\int_{}^{}1.dx=x+c$","ts":1600255089431,"cs":"wAD3ca6Sq5N5AI0P+IZsIw==","size":{"width":113,"height":20}}

{"code":"\\begin{align*}\n{\\int_{}^{}f\\left(x\\right)dx}&={\\frac{1}{2}\\left[-\\log_{}\\left|t\\right|+x\\right]+C}\t\n\\end{align*}","id":"2-1-1-0-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-2-1-1-1-0","font":{"size":10,"family":"Arial","color":"#000000"},"type":"align*","ts":1600255149223,"cs":"qD1rPIhVzih/E38lGDLEAQ==","size":{"width":221,"height":36}}

Put the value of t:-

{"code":"\\begin{align*}\n{\\int_{}^{}f\\left(x\\right)dx}&={\\frac{1}{2}\\left(-\\log_{}\\left|\\cos x-\\sin x\\right|+x\\right)+C}\\\\\n{\\int_{}^{}f\\left(x\\right)dx}&={\\frac{x}{2}-\\frac{1}{2}\\log_{}\\left|\\cos x-\\sin x\\right|+C}\t\n\\end{align*}","font":{"size":10,"family":"Arial","color":"#000000"},"type":"align*","id":"2-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-2-1-1-1-0","ts":1600256919553,"cs":"CMMghIm0TafwU00GCVhXHg==","size":{"width":302,"height":76}}

{"code":"$34.\\,\\frac{{\\sqrt[]{\\tan x}}}{\\sin x.\\cos x}$","font":{"color":"#000000","family":"Arial","size":12},"type":"$","id":"1-2-1-1-1-1-1-1-1-1-1-1-1-1-1-2-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-0","ts":1600256992535,"cs":"DQBCaVWQRDnH8ZIEV0ax2A==","size":{"width":100,"height":28}}

Solun:- Let f(x) = {"type":"$","code":"$\\frac{{\\sqrt[]{\\tan x}}}{\\sin x.\\cos x}$","id":"11-1-1-1-1-1-0","font":{"size":11.666666666666668,"color":"#000000","family":"Arial"},"ts":1600257105347,"cs":"EEmW7VRVOc3mtuVcIsi19A==","size":{"width":70,"height":28}}

Integrate f(x):-

{"code":"\\begin{align*}\n{\\int_{}^{}f\\left(x\\right)dx}&={\\int_{}^{}\\frac{{\\sqrt[]{\\tan x}}}{\\sin x.\\cos x}.dx}\\\\\n{\\int_{}^{}f\\left(x\\right)dx}&={\\int_{}^{}\\frac{{\\sqrt[]{\\tan x}}.\\cos x}{\\sin x.\\cos ^{2}x}.dx}\\\\\n{\\int_{}^{}f\\left(x\\right)dx}&={\\int_{}^{}\\frac{{\\sqrt[]{\\tan x}}}{\\frac{\\sin x}{\\cos x}.\\cos ^{2}x}.dx}\\\\\n{\\int_{}^{}f\\left(x\\right)dx}&={\\int_{}^{}\\frac{\\sec^{2}x.{\\sqrt[]{\\tan x}}}{\\tan x}.dx}\\\\\n{\\int_{}^{}f\\left(x\\right)dx}&={\\int_{}^{}\\frac{\\sec^{2}x}{{\\sqrt[]{\\tan x}}}.dx}\t\n\\end{align*}","type":"align*","font":{"color":"#000000","size":10,"family":"Arial"},"id":"2-0-1-1-1-1-1-1-1-1-1-1-2-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-0","ts":1600257714545,"cs":"fhDsHpEPbO2gYWu9FNSqrg==","size":{"width":232,"height":221}}

Let tan x = t2

Differentiate w.r.t to t:-

⇒ sec2x.dx = 2t.dt

{"type":"align*","id":"2-1-0-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-2-1-1-1-0-1-1-1-1-1-1-1-2-1-1-1-1-0","code":"\\begin{align*}\n{\\int_{}^{}f\\left(x\\right)dx}&={\\int_{}^{}\\frac{2t.dt}{t}}\\\\\n{\\int_{}^{}f\\left(x\\right)dx}&={\\int_{}^{}2.dt}\t\n\\end{align*}","font":{"color":"#000000","size":10,"family":"Arial"},"ts":1600258128848,"cs":"OM2xxqi+e1hLNGovNvqfVA==","size":{"width":148,"height":76}}

We know that:-

{"code":"$\\int_{}^{}1.dx=x+c$","id":"3-1-2-1-1-4-1-4-1-1-1-1-2-1-1-1-1-1-1-1-2-1-1-0-2-0","type":"$","font":{"size":11,"family":"Arial","color":"#000000"},"ts":1600258217330,"cs":"k3XufrITymC4kFGd+cHqxg==","size":{"width":113,"height":20}}

{"font":{"family":"Arial","color":"#000000","size":10},"id":"2-1-1-0-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-2-1-1-1-1-0","code":"\\begin{align*}\n{\\int_{}^{}f\\left(x\\right)dx}&={2t+C}\t\n\\end{align*}","type":"align*","ts":1600258250873,"cs":"v1jmsu8mW16dw/gO2QiVWg==","size":{"width":132,"height":36}}

Put the value of t:-

{"code":"\\begin{align*}\n{\\int_{}^{}f\\left(x\\right)dx}&={2{\\sqrt[]{\\tan x}}+C}\t\n\\end{align*}","font":{"size":10,"color":"#000000","family":"Arial"},"type":"align*","id":"2-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-2-1-1-1-1-0","ts":1600257982543,"cs":"y2I157X7BgVAP9fGCgY2rQ==","size":{"width":174,"height":36}}

{"id":"1-2-1-1-1-1-1-1-1-1-1-1-1-1-1-2-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-0","type":"$","font":{"color":"#000000","family":"Arial","size":12},"code":"$35.\\,\\frac{\\left(1+\\log_{}x\\right)^{2}}{x}$","ts":1600259942029,"cs":"X29ygOnn/YAvG/RW99N8xg==","size":{"width":104,"height":30}}

Solun:- Let f(x) = {"type":"$","code":"$\\frac{\\left(1+\\log_{}x\\right)^{2}}{x}$","font":{"color":"#000000","size":11.666666666666668,"family":"Arial"},"id":"11-1-1-1-1-1-1-0","ts":1600259966903,"cs":"e1fRKHzjkS7mtGAj97uXsQ==","size":{"width":73,"height":30}}

Integrate f(x):-

{"id":"2-0-1-1-1-1-1-1-1-1-1-1-2-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-0","type":"align*","font":{"color":"#000000","size":10,"family":"Arial"},"code":"\\begin{align*}\n{\\int_{}^{}f\\left(x\\right)dx}&={\\int_{}^{}\\frac{\\left(1+\\log_{}x\\right)^{2}}{x}.dx}\t\n\\end{align*}","ts":1600260037304,"cs":"aIo6TEoCHdfSd//JRmFHbw==","size":{"width":216,"height":40}}

Let (1 + log x) = t

Differentiate w.r.t to t:-

⇒ 1/x.dx = dt

{"font":{"color":"#000000","size":10,"family":"Arial"},"code":"\\begin{align*}\n{\\int_{}^{}f\\left(x\\right)dx}&={\\int_{}^{}t^{2}.dt}\t\n\\end{align*}","type":"align*","id":"2-1-0-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-2-1-1-1-0-1-1-1-1-1-1-1-2-1-1-1-1-1-0","ts":1600260116257,"cs":"MAi/eRFfO9D26of5Z/ps6Q==","size":{"width":140,"height":36}}

We know that:-

{"type":"$","font":{"size":11,"family":"Arial","color":"#000000"},"code":"$\\int_{}^{}x^{n}.dx=\\frac{x^{n+1}}{n+1}+c$","id":"3-1-2-1-1-4-1-4-1-1-1-1-2-1-1-1-1-1-1-1-2-1-1-0-2-1-0","ts":1600260172724,"cs":"fq3Wn7DLOCKXAZ8C1Jsspw==","size":{"width":149,"height":24}}

{"font":{"size":10,"family":"Arial","color":"#000000"},"id":"2-1-1-0-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-2-1-1-1-1-1-0","type":"align*","code":"\\begin{align*}\n{\\int_{}^{}f\\left(x\\right)dx}&={\\frac{t^{3}}{3}+C}\t\n\\end{align*}","ts":1600260207419,"cs":"CIuNe9ZF3dfNzz2M2qJ8yA==","size":{"width":137,"height":37}}

Put the value of t:-

{"font":{"size":10,"family":"Arial","color":"#000000"},"id":"2-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-2-1-1-1-1-1-0","type":"align*","code":"\\begin{align*}\n{\\int_{}^{}f\\left(x\\right)dx}&={\\frac{\\left(1+\\log_{}x\\right)^{3}}{3}+C}\t\n\\end{align*}","ts":1600260529332,"cs":"uwHXCL+8sa3HGpbBmWbulQ==","size":{"width":204,"height":40}}

{"id":"1-2-1-1-1-1-1-1-1-1-1-1-1-1-1-2-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-0","font":{"color":"#000000","family":"Arial","size":12},"code":"$36.\\,\\frac{\\left(x+1\\right)\\left(x+\\log_{}x\\right)^{2}}{x}$","type":"$","ts":1600260726848,"cs":"nBk4eK8gQOVCJdVWvdgmig==","size":{"width":141,"height":30}}

Solun:- Let f(x) = {"code":"$\\frac{\\left(x+1\\right)\\left(x+\\log_{}x\\right)^{2}}{x}$","font":{"size":11.666666666666668,"family":"Arial","color":"#000000"},"type":"$","id":"11-1-1-1-1-1-1-1-0","ts":1600260754922,"cs":"AhhiuQVmhCo3REeAqN8dfQ==","size":{"width":112,"height":30}}

Integrate f(x):-

{"code":"\\begin{align*}\n{\\int_{}^{}f\\left(x\\right)dx}&={\\int_{}^{}\\frac{\\left(x+1\\right)\\left(x+\\log_{}x\\right)^{2}}{x}.dx}\\\\\n{\\int_{}^{}f\\left(x\\right)dx}&={\\int_{}^{}\\frac{\\left(x+1\\right)}{x}\\left(x+\\log_{}x\\right)^{2}.dx}\\\\\n{\\int_{}^{}f\\left(x\\right)dx}&={\\int_{}^{}\\left(1+\\frac{1}{x}\\right)\\left(x+\\log_{}x\\right)^{2}.dx}\t\n\\end{align*}","type":"align*","font":{"color":"#000000","family":"Arial","size":10},"id":"2-0-1-1-1-1-1-1-1-1-1-1-2-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-0","ts":1600260929185,"cs":"BIRTtT8ZW4/RJcydgPSAIA==","size":{"width":276,"height":124}}

Let (x + log x) = t

Differentiate w.r.t to t:-

⇒ (1+1/x).dx = dt

{"font":{"color":"#000000","size":10,"family":"Arial"},"code":"\\begin{align*}\n{\\int_{}^{}f\\left(x\\right)dx}&={\\int_{}^{}t^{2}.dt}\t\n\\end{align*}","type":"align*","id":"2-1-0-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-2-1-1-1-0-1-1-1-1-1-1-1-2-1-1-1-1-1-1-0","ts":1600260116257,"cs":"Pm2V6+gwBTrfSDtOuzrRig==","size":{"width":140,"height":36}}

We know that:-

{"type":"$","font":{"size":11,"family":"Arial","color":"#000000"},"code":"$\\int_{}^{}x^{n}.dx=\\frac{x^{n+1}}{n+1}+c$","id":"3-1-2-1-1-4-1-4-1-1-1-1-2-1-1-1-1-1-1-1-2-1-1-0-2-1-1-0","ts":1600260172724,"cs":"8OQ/SJz1cq9kku3+kNx57g==","size":{"width":149,"height":24}}

{"font":{"size":10,"family":"Arial","color":"#000000"},"id":"2-1-1-0-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-2-1-1-1-1-1-1","type":"align*","code":"\\begin{align*}\n{\\int_{}^{}f\\left(x\\right)dx}&={\\frac{t^{3}}{3}+C}\t\n\\end{align*}","ts":1600260207419,"cs":"rq1zdnSbNQrMw2S+Z6kIkQ==","size":{"width":137,"height":37}}

Put the value of t:-

{"id":"2-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-2-1-1-1-1-1-1-0","type":"align*","code":"\\begin{align*}\n{\\int_{}^{}f\\left(x\\right)dx}&={\\frac{\\left(x+\\log_{}x\\right)^{3}}{3}+C}\t\n\\end{align*}","font":{"size":10,"family":"Arial","color":"#000000"},"ts":1600261143375,"cs":"GOAnpbu/gdHNyBNfiqYLow==","size":{"width":205,"height":40}}

{"id":"1-2-1-1-1-1-1-1-1-1-1-1-1-1-1-2-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1","code":"$37.\\,\\frac{x^{3}\\sin\\left(\\tan^{-1}x^{4}\\right)}{1+x^{8}}$","type":"$","font":{"color":"#000000","size":12,"family":"Arial"},"ts":1600261616943,"cs":"wex1j2BIv0Wlp9xGLLONIA==","size":{"width":136,"height":32}}

Solun:- Let f(x) = {"type":"$","code":"$\\frac{x^{3}\\sin\\left(\\tan^{-1}x^{4}\\right)}{1+x^{8}}$","id":"11-1-1-1-1-1-1-1-1","font":{"color":"#000000","family":"Arial","size":11.666666666666668},"ts":1600261587313,"cs":"h4FFxpvjjTHvNWZctJKg9w==","size":{"width":106,"height":32}}

Integrate f(x):-

{"id":"2-0-1-1-1-1-1-1-1-1-1-1-2-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1","type":"align*","font":{"family":"Arial","color":"#000000","size":10},"code":"\\begin{align*}\n{\\int_{}^{}f\\left(x\\right)dx}&={\\int_{}^{}\\frac{x^{3}\\sin\\left(\\tan^{-1}x^{4}\\right)}{1+x^{8}}.dx}\t\n\\end{align*}","ts":1600261565081,"cs":"PJcCLWwqi+q41LZ+afFj+w==","size":{"width":246,"height":40}}

Let tan-1(x)4 = t

Differentiate w.r.t to t:-

{"code":"\\begin{align*}\n{\\frac{1}{1+\\left(x^{4}\\right)^{2}}\\times4x^{3}.dx}&={dt}\\\\\n{\\frac{1}{1+\\left(x^{4}\\right)^{2}}\\times \\,x^{3}.dx}&={\\frac{1}{4}.dt}\\\\\n{\\frac{x^{3}}{1+\\left(x^{8}\\right)}\\,.\\,dx\\,\\,\\,\\,\\,\\,\\,\\,\\,}&={\\frac{1}{4}.dt}\t\n\\end{align*}","id":"12-0","type":"align*","font":{"size":10,"family":"Arial","color":"#000000"},"ts":1600262044698,"cs":"t29HYu1ZYYsWiV8XBjN4cw==","size":{"width":192,"height":128}}

{"font":{"family":"Arial","size":10,"color":"#000000"},"type":"align*","code":"\\begin{align*}\n{\\int_{}^{}f\\left(x\\right)dx}&={\\frac{1}{4}\\int_{}^{}\\sin t.dt}\t\n\\end{align*}","id":"2-1-0-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-2-1-1-1-0-1-1-1-1-1-1-1-2-1-1-1-1-1-1-1-0","ts":1600262334146,"cs":"bh/fORmhXVmimOtxqVHHuA==","size":{"width":173,"height":36}}

We know that:-

{"code":"$\\int_{}^{}\\sin x.dx=-\\cos x+c$","id":"3-1-2-1-1-4-1-4-1-1-1-1-2-1-1-1-1-1-1-1-2-1-1-0-2-1-1-1-0","font":{"family":"Arial","size":11,"color":"#000000"},"type":"$","ts":1600262166022,"cs":"DX0pqRK3ZMjAl9uLn2f8yg==","size":{"width":188,"height":20}}

{"font":{"family":"Arial","color":"#000000","size":10},"type":"align*","id":"2-1-1-0-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-2-1-1-1-1-1-2-0","code":"\\begin{align*}\n{\\int_{}^{}f\\left(x\\right)dx}&={\\frac{-1}{4}\\cos t+C}\t\n\\end{align*}","ts":1600262399486,"cs":"+1S+4UNW79arIFUJxPKfMw==","size":{"width":176,"height":36}}

Put the value of t:-

{"font":{"family":"Arial","color":"#000000","size":10},"id":"2-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-2-1-1-1-1-1-1-1-0","code":"\\begin{align*}\n{\\int_{}^{}f\\left(x\\right)dx}&={\\frac{-1}{4}\\cos\\left(\\tan^{-1}x^{4}\\right)+C}\t\n\\end{align*}","type":"align*","ts":1600262440527,"cs":"Xb4PHoF1AOBvJk8KCZCfYw==","size":{"width":237,"height":36}}

Choose the correct answer in Exercises 38 and 39.

{"id":"1-0-1-0","code":"$38.\\,\\int_{}^{}\\frac{10x^{9}+10^{x}\\log_{e}10}{x^{10}+10^{x}}dx$","font":{"color":"#000000","size":12,"family":"Arial"},"type":"$","ts":1600334748914,"cs":"h/st0nfCf+s9N4KkJX5efA==","size":{"width":185,"height":30}}

Solun:- Given

{"id":"13-0","type":"$","font":{"color":"#000000","size":12,"family":"Arial"},"code":"$=\\int_{}^{}\\frac{10x^{9}+10^{x}\\log_{e}10}{x^{10}+10^{x}}dx$","ts":1600335720156,"cs":"QZkuC+zwQMSot48u5d/jZw==","size":{"width":173,"height":30}}

Let x10+10x = t

Differentiate w.r.t to t:-

{"code":"\\begin{align*}\n{\\left(10x^{9}+10^{x}\\log_{e}10\\right)dx}&={dt}\t\n\\end{align*}","font":{"color":"#000000","size":10,"family":"Arial"},"id":"12-1-0","type":"align*","ts":1600335644001,"cs":"i1gDE2sae/Num6TqspiMkg==","size":{"width":188,"height":20}}

{"type":"align*","font":{"size":10,"color":"#000000","family":"Arial"},"code":"\\begin{align*}\n{}&={\\int_{}^{}\\frac{dt}{t}}\t\n\\end{align*}","id":"2-1-0-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-2-1-1-1-0-1-1-1-1-1-1-1-2-1-1-1-1-1-1-1-1-0","ts":1600335700977,"cs":"NsyKi/Rv7H0RuE5xvVATfg==","size":{"width":61,"height":36}}

We know that:-

{"id":"3-1-2-1-1-4-1-4-1-1-1-1-2-1-1-1-1-1-1-1-2-1-1-0-2-1-1-1-1-0","font":{"family":"Arial","color":"#000000","size":11},"type":"$","code":"$\\int_{}^{}\\frac{1}{x}.dx=\\log_{}\\left|x\\right|+c$","ts":1600335758781,"cs":"1dtWcHdDmxpD+zmIGe8Zsw==","size":{"width":157,"height":21}}

{"code":"\\begin{align*}\n{}&={\\log_{}t+C}\t\n\\end{align*}","font":{"family":"Arial","size":10,"color":"#000000"},"id":"2-1-1-0-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-2-1-1-1-1-1-2-1-0","type":"align*","ts":1600335805477,"cs":"sKJHSxc7kUn2md9KAH97Ow==","size":{"width":81,"height":16}}

Put the value of t:-

{"font":{"color":"#000000","size":10,"family":"Arial"},"code":"\\begin{align*}\n{\\int_{}^{}f\\left(x\\right)dx}&={\\log_{}\\left|x^{10}+10^{x}\\right|+C}\t\n\\end{align*}","id":"2-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-2-1-1-1-1-1-1-1-1-0","type":"align*","ts":1600335836327,"cs":"mC45Q8N8jaHS4Ed/Qwmd7A==","size":{"width":213,"height":36}}

The correct answer is D.

{"type":"$","code":"$39.\\,\\int_{}^{}\\frac{1.dx}{\\sin^{2}x.\\cos^{2}x}$","id":"1-0-1-1","font":{"color":"#000000","family":"Arial","size":12},"ts":1600335970965,"cs":"9JYFb7DjT7CaETKp5shBCg==","size":{"width":133,"height":26}}

Solun:- Given

{"id":"13-1","font":{"color":"#000000","family":"Arial","size":10},"code":"\\begin{align*}\n{}&={\\int_{}^{}\\frac{1.dx}{\\sin^{2}x.\\cos^{2}x}}\\\\\n{}&={\\int_{}^{}\\frac{1.dx}{\\frac{\\sin^{2}x.\\cos^{2}x.\\cos^{2}x}{\\cos^{2}x}}}\\\\\n{}&={\\int_{}^{}\\frac{\\sec^{2}x.dx}{\\tan^{2}x.\\cos^{2}x}}\\\\\n{}&={\\int_{}^{}\\frac{\\sec^{2}x.\\sec^{2}x.dx}{\\tan^{2}x}}\\\\\n{}&={\\int_{}^{}\\frac{\\left(1+\\tan^{2}x\\right).\\sec^{2}x.dx}{\\tan^{2}x}}\t\n\\end{align*}","type":"align*","ts":1600336880667,"cs":"wFX5JX53zEb9YqFd4uM3IQ==","size":{"width":200,"height":214}}

Let tan x = t

Differentiate w.r.t to t:-

⇒ sec2 x.dx = dt

{"font":{"color":"#000000","family":"Arial","size":10},"id":"2-1-0-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-2-1-1-1-0-1-1-1-1-1-1-1-2-1-1-1-1-1-1-1-1-1","type":"align*","code":"\\begin{align*}\n{}&={\\int_{}^{}\\frac{\\left(1+t^{2}\\right).dt}{t^{2}}}\\\\\n{}&={\\int_{}^{}\\frac{1.dt}{t^{2}}+\\int_{}^{}\\frac{t^{2}.dt}{t^{2}}}\\\\\n{}&={\\int_{}^{}t^{-2}.dt+\\int_{}^{}1.dt}\t\n\\end{align*}","ts":1600337067646,"cs":"ddiP2mH3spqZ2TxN6RYWHQ==","size":{"width":152,"height":124}}

We know that:-

{"id":"3-1-2-1-1-4-1-4-1-1-1-1-2-1-1-1-1-1-1-1-2-1-1-0-2-1-1-1-1-1-0","font":{"color":"#000000","size":11,"family":"Arial"},"type":"$","code":"$\\int_{}^{}x^{n}.dx=\\frac{x^{n+1}}{n+1}+c$","ts":1600337152930,"cs":"7kxeMPs7G1I13fUnKilfgA==","size":{"width":149,"height":24}}

{"id":"3-1-2-1-1-4-1-4-1-1-1-1-2-1-1-1-1-1-1-1-2-1-1-0-2-1-1-1-1-1-1","code":"$\\int_{}^{}1.dx=x+c$","font":{"size":11,"color":"#000000","family":"Arial"},"type":"$","ts":1600337246573,"cs":"VtAVLv6veeKB9XqW2ClwpQ==","size":{"width":113,"height":20}}

{"font":{"color":"#000000","family":"Arial","size":10},"code":"\\begin{align*}\n{}&={\\frac{t^{-1}}{-1}+t+C}\\\\\n{}&={\\frac{-1}{t}+t+C}\t\n\\end{align*}","id":"2-1-1-0-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-2-1-1-1-1-1-2-1-1","type":"align*","ts":1600337525115,"cs":"XpCNvLpcD848RAO1p5FoaQ==","size":{"width":104,"height":72}}

Put the value of t:-

{"type":"align*","id":"2-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-2-1-1-1-1-1-1-1-1-1","code":"\\begin{align*}\n{}&={\\frac{-1}{\\tan x}+\\tan x+C}\\\\\n{}&={\\tan x-\\cot x+C}\t\n\\end{align*}","font":{"color":"#000000","size":10,"family":"Arial"},"ts":1600337425985,"cs":"lItimPe5RSqW0ZDDhIu4Iw==","size":{"width":148,"height":52}}

The correct answer is B.


Download PDF of Exercise 7.2

See Also:-

Notes of Integrals

Exercise 7.1


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