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Exercise 7.11

By using the properties of definite integrals, evaluate the integrals in Exercises 1 to 19.

{"font":{"size":10,"family":"Arial","color":"#000000"},"id":"4-0-0-0","type":"$","code":"$1.\\,\\int_{0}^{\\frac{\\Pi}{2}}\\cos^{2}x.dx$","ts":1602403399921,"cs":"gA/RsPP8nL9t2jangjYOiA==","size":{"width":105,"height":24}}

Solun:- Let f(x) = cos2x

Integrate f(x):-

{"font":{"color":"#000000","family":"Arial","size":10},"id":"2-0-0-0-0-0-0-0-0-0-0-0-0-0-0-0","type":"align*","code":"\\begin{align*}\n{I}&={\\int_{0}^{\\frac{\\Pi}{2}}\\cos^{2}x.dx....\\left(1\\right)}\\\\\n{I}&={\\int_{0}^{\\frac{\\Pi}{2}}\\left[\\cos x\\right]^{2}.dx}\t\n\\end{align*}","ts":1602404001899,"cs":"ZcBbDgkPJyVKkxsl4HRqDA==","size":{"width":170,"height":88}}

By Property:-

{"font":{"family":"Arial","color":"#000000","size":10},"type":"$","id":"1-0","code":"$\\int_{a}^{b}f\\left(x\\right).dx=\\int_{a}^{b}f\\left(a+b-x\\right).dx$","ts":1602403552089,"cs":"oSLWbWKUzCnbBbKPEPQTtQ==","size":{"width":220,"height":21}}

{"code":"\\begin{align*}\n{I}&={\\int_{0}^{\\frac{\\Pi}{2}}\\left[\\cos\\left(\\frac{\\Pi}{2}-x\\right)\\right]^{2}.dx}\\\\\n{I}&={\\int_{0}^{\\frac{\\Pi}{2}}\\left[\\sin x\\right]^{2}.dx}\\\\\n{I}&={\\int_{0}^{\\frac{\\Pi}{2}}\\sin ^{2}x.dx....\\left(2\\right)}\t\n\\end{align*}","type":"align*","id":"2-0-0-0-0-0-0-0-0-0-0-0-0-0-1-0","font":{"size":10,"color":"#000000","family":"Arial"},"ts":1602404510961,"cs":"/FbeHpqoeRKInJQN5i5qTw==","size":{"width":200,"height":134}}

Add eq. 1 and 2:-

{"id":"3-0-0-0-0-0-0-1-0-0-0-0-0-0","font":{"color":"#000000","family":"Arial","size":10},"code":"\\begin{align*}\n{2I}&={\\int_{0}^{\\frac{\\Pi}{2}} \\cos^{2}x.dx+\\int_{0}^{\\frac{\\Pi}{2}}\\sin ^{2}x.dx}\\\\\n{2I}&={\\int_{0}^{\\frac{\\Pi}{2}}\\cos^{2}x+\\sin ^{2}x.dx}\\\\\n{2I}&={\\int_{0}^{\\frac{\\Pi}{2}}1.dx}\\\\\n{2I}&={\\left[x\\right]_{0}^{\\frac{\\Pi}{2}}}\\\\\n{2I}&={\\frac{\\Pi}{2}}\\\\\n{I\\,\\,}&={\\frac{\\Pi}{4}}\t\n\\end{align*}","type":"align*","ts":1602404225546,"cs":"aqfYBdplKyKgR9NO/yAP+A==","size":{"width":242,"height":236}}

{"id":"4-0-0-1-0-0-0","font":{"color":"#000000","family":"Arial","size":10},"type":"$","code":"$2.\\,\\int_{0}^{\\frac{\\Pi}{2}}\\frac{{\\sqrt[]{\\sin x}}}{{\\sqrt[]{\\sin x}}+{\\sqrt[]{\\cos x}}}.dx$","ts":1602404307253,"cs":"ZyIsaxIH3NJIQRztgUCM2Q==","size":{"width":146,"height":29}}

Solun:- Let f(x) = {"font":{"color":"#000000","size":10,"family":"Arial"},"code":"$\\frac{{\\sqrt[]{\\sin x}}}{{\\sqrt[]{\\sin x}}+{\\sqrt[]{\\cos x}}}$","type":"$","id":"15-0","ts":1602404331047,"cs":"iCZ0LsQnhtqIFBat0Vny1Q==","size":{"width":80,"height":28}}

Integrate f(x):-

{"id":"2-0-0-0-0-0-0-0-0-0-0-0-0-0-0-1-0-0-0","code":"\\begin{align*}\n{I}&={\\int_{0}^{\\frac{\\Pi}{2}}\\frac{{\\sqrt[]{\\sin x}}}{{\\sqrt[]{\\sin x}}+{\\sqrt[]{\\cos x}}}.dx....\\left(1\\right)}\t\n\\end{align*}","type":"align*","font":{"color":"#000000","family":"Arial","size":10},"ts":1602404353715,"cs":"I+fy9+8C7J/3r2WN0y6Q0A==","size":{"width":249,"height":44}}

By Property:-

{"font":{"family":"Arial","color":"#000000","size":10},"type":"$","id":"1-1","code":"$\\int_{a}^{b}f\\left(x\\right).dx=\\int_{a}^{b}f\\left(a+b-x\\right).dx$","ts":1602403552089,"cs":"bbUey+1BLemSHcefOSC9vQ==","size":{"width":220,"height":21}}

{"code":"\\begin{align*}\n{I}&={\\int_{0}^{\\frac{\\Pi}{2}}\\frac{{\\sqrt[]{\\sin x}}}{{\\sqrt[]{\\sin x}}+{\\sqrt[]{\\cos x}}}.dx}\\\\\n{I}&={\\int_{0}^{\\frac{\\Pi}{2}}\\frac{{\\sqrt[]{\\sin \\left(\\frac{\\Pi}{2}-x\\right)}}}{{\\sqrt[]{\\sin \\left(\\frac{\\Pi}{2}-x\\right)}}+{\\sqrt[]{\\cos \\left(\\frac{\\Pi}{2}-x\\right)}}}.dx}\\\\\n{I}&={\\int_{0}^{\\frac{\\Pi}{2}}\\frac{{\\sqrt[]{\\cos x}}}{{\\sqrt[]{\\cos x}}+{\\sqrt[]{\\sin x}}}.dx....\\left(2\\right)}\t\n\\end{align*}","type":"align*","font":{"size":10,"color":"#000000","family":"Arial"},"id":"2-0-0-0-0-0-0-0-0-0-0-0-0-0-1-1-0-0-0","ts":1602404533810,"cs":"0Joyhgfcp7Bw/Rfac4tClw==","size":{"width":304,"height":164}}

Add eq. 1 and 2:-

{"type":"align*","font":{"family":"Arial","size":10,"color":"#000000"},"code":"\\begin{align*}\n{2I}&={\\int_{0}^{\\frac{\\Pi}{2}}\\frac{{\\sqrt[]{\\cos x}}}{{\\sqrt[]{\\cos x}}+{\\sqrt[]{\\sin x}}}.dx+\\int_{0}^{\\frac{\\Pi}{2}}\\frac{{\\sqrt[]{\\sin x}}}{{\\sqrt[]{\\cos x}}+{\\sqrt[]{\\sin x}}}.dx}\\\\\n{2I}&={\\int_{0}^{\\frac{\\Pi}{2}}\\frac{{\\sqrt[]{\\cos x}}+{\\sqrt[]{\\sin x}}}{{\\sqrt[]{\\cos x}}+{\\sqrt[]{\\sin x}}}.dx}\\\\\n{2I}&={\\int_{0}^{\\frac{\\Pi}{2}}1.dx}\\\\\n{2I}&={\\left[x\\right]_{0}^{\\frac{\\Pi}{2}}}\\\\\n{2I}&={\\frac{\\Pi}{2}}\\\\\n{I\\,\\,}&={\\frac{\\Pi}{4}}\t\n\\end{align*}","id":"3-0-0-0-0-0-0-1-0-0-0-0-0-1-0-0-0","ts":1602404610325,"cs":"PfeDMwoS/5HbnhjVIaMF9w==","size":{"width":400,"height":244}}

{"code":"$3.\\,\\int_{0}^{\\frac{\\Pi}{2}}\\frac{\\sin ^{\\frac{3}{2}}x}{\\sin ^{\\frac{3}{2}}x+ \\cos^{\\frac{3}{2}}x}.dx$","id":"4-0-0-1-1-0","font":{"color":"#000000","family":"Arial","size":10},"type":"$","ts":1602405385609,"cs":"eayIOFgljk1dZRZ0jyq8HQ==","size":{"width":144,"height":32}}

Solun:- Let f(x) = {"code":"$\\frac{\\sin ^{\\frac{3}{2}}x}{\\sin ^{\\frac{3}{2}}x+ \\cos^{\\frac{3}{2}}x}$","id":"5","type":"$","font":{"size":10,"family":"Arial","color":"#000000"},"ts":1602405421404,"cs":"QapBMrjJaisLqdkG4Rnh4w==","size":{"width":76,"height":32}}

Integrate f(x):-

{"font":{"size":10,"family":"Arial","color":"#000000"},"type":"align*","id":"2-0-0-0-0-0-0-0-0-0-0-0-0-0-0-1-1-0","code":"\\begin{align*}\n{I}&={\\int_{0}^{\\frac{\\Pi}{2}}\\frac{\\sin ^{\\frac{3}{2}}x}{\\sin ^{\\frac{3}{2}}x+ \\cos^{\\frac{3}{2}}x}.dx....\\left(1\\right)}\\\\\n{I}&={\\int_{0}^{\\frac{\\Pi}{2}}\\frac{\\left[\\sin x\\right]^{\\frac{3}{2}}}{\\left[\\sin x\\right]^{\\frac{3}{2}}+ \\left[\\cos x\\right]^{\\frac{3}{2}}}.dx}\t\n\\end{align*}","ts":1602405534440,"cs":"0fgHm/GgnCs5WSuoDWW0Mw==","size":{"width":241,"height":96}}

By Property:-

{"font":{"family":"Arial","color":"#000000","size":10},"type":"$","id":"1-2-0","code":"$\\int_{a}^{b}f\\left(x\\right).dx=\\int_{a}^{b}f\\left(a+b-x\\right).dx$","ts":1602403552089,"cs":"gslxqq6s+yyfg78P1/DHEQ==","size":{"width":220,"height":21}}

{"type":"align*","id":"2-0-0-0-0-0-0-0-0-0-0-0-0-0-1-1-1-0","code":"\\begin{align*}\n{I}&={\\int_{0}^{\\frac{\\Pi}{2}}\\frac{\\left[\\sin \\left(\\frac{\\Pi}{2}-x\\right)\\right]^{\\frac{3}{2}}}{\\left[\\sin \\left(\\frac{\\Pi}{2}-x\\right)\\right]^{\\frac{3}{2}}+ \\left[\\cos \\left(\\frac{\\Pi}{2}-x\\right)\\right]^{\\frac{3}{2}}}.dx}\\\\\n{I}&={\\int_{0}^{\\frac{\\Pi}{2}}\\frac{\\left[\\cos x\\right]^{\\frac{3}{2}}}{\\left[\\cos x\\right]^{\\frac{3}{2}}+ \\left[\\sin x\\right]^{\\frac{3}{2}}}.dx}\\\\\n{I}&={\\int_{0}^{\\frac{\\Pi}{2}}\\frac{\\cos^{\\frac{3}{2}}x}{\\cos^{\\frac{3}{2}}x+\\sin^{\\frac{3}{2}}x}.dx....\\left(2\\right)}\t\n\\end{align*}","font":{"color":"#000000","family":"Arial","size":10},"ts":1602492607270,"cs":"/t8Lb8QwDo+04BRDUPZgBw==","size":{"width":317,"height":157}}

Add eq. 1 and 2:-

{"font":{"size":10,"color":"#000000","family":"Arial"},"id":"3-0-0-0-0-0-0-1-0-0-0-0-0-1-1","code":"\\begin{align*}\n{2I}&={\\int_{0}^{\\frac{\\Pi}{2}}\\frac{\\cos^{\\frac{3}{2}}x}{\\cos^{\\frac{3}{2}}x+\\sin^{\\frac{3}{2}}x}.dx+\\int_{0}^{\\frac{\\Pi}{2}}\\frac{\\sin^{\\frac{3}{2}}x}{\\cos^{\\frac{3}{2}}x+\\sin^{\\frac{3}{2}}x}.dx}\\\\\n{2I}&={\\int_{0}^{\\frac{\\Pi}{2}}\\frac{\\sin^{\\frac{3}{2}}x+\\cos^{\\frac{3}{2}}x}{\\cos^{\\frac{3}{2}}x+\\sin^{\\frac{3}{2}}x}.dx}\\\\\n{2I}&={\\int_{0}^{\\frac{\\Pi}{2}}1.dx}\\\\\n{2I}&={\\left[x\\right]_{0}^{\\frac{\\Pi}{2}}}\\\\\n{2I}&={\\frac{\\Pi}{2}}\\\\\n{I\\,\\,}&={\\frac{\\Pi}{4}}\t\n\\end{align*}","type":"align*","ts":1602492824977,"cs":"/A9bTOdPoZ0TWoMlUqnvug==","size":{"width":384,"height":240}}

{"type":"$","id":"4-0-0-1-1-1-0-0","font":{"color":"#000000","size":10,"family":"Arial"},"code":"$4.\\,\\int_{0}^{\\frac{\\Pi}{2}}\\frac{\\cos^{5}x}{\\sin ^{5}x+ \\cos^{5}x}.dx$","ts":1602492987430,"cs":"Bdu1JLftVsVGYpDrLkDkYQ==","size":{"width":136,"height":26}}

Solun:- Let f(x) = {"id":"6-0","type":"$","font":{"family":"Arial","size":10,"color":"#000000"},"code":"$\\frac{\\cos^{5}x}{\\sin ^{5}x+ \\cos^{5}x}$","ts":1602493037893,"cs":"P0/xteOKWWuB1y3KPsdi9g==","size":{"width":69,"height":24}}

Integrate f(x):-

{"font":{"color":"#000000","family":"Arial","size":10},"id":"2-0-0-0-0-0-0-0-0-0-0-0-0-0-0-1-1-1-0-0","code":"\\begin{align*}\n{I}&={\\int_{0}^{\\frac{\\Pi}{2}}\\frac{\\cos^{5}x}{\\sin ^{5}x+ \\cos^{5}x}.dx....\\left(1\\right)}\\\\\n{I}&={\\int_{0}^{\\frac{\\Pi}{2}}\\frac{\\left[\\cos x\\right]^{5}}{\\left[\\sin x\\right]^{5}+ \\left[\\cos x\\right]^{5}}.dx}\t\n\\end{align*}","type":"align*","ts":1602493092402,"cs":"VHMqHI1rbTtUPfjsT28cCA==","size":{"width":234,"height":92}}

By Property:-

{"font":{"family":"Arial","color":"#000000","size":10},"type":"$","id":"1-2-1-0","code":"$\\int_{a}^{b}f\\left(x\\right).dx=\\int_{a}^{b}f\\left(a+b-x\\right).dx$","ts":1602403552089,"cs":"JXH77grhfHOnbDP+0K3WaQ==","size":{"width":220,"height":21}}

{"font":{"family":"Arial","size":10,"color":"#000000"},"type":"align*","id":"2-0-0-0-0-0-0-0-0-0-0-0-0-0-1-1-1-1-0-0-0","code":"\\begin{align*}\n{I}&={\\int_{0}^{\\frac{\\Pi}{2}}\\frac{\\left[\\cos \\left(\\frac{\\Pi}{2}-x\\right)\\right]^{5}}{\\left[\\sin \\left(\\frac{\\Pi}{2}-x\\right)\\right]^{5}+ \\left[\\cos \\left(\\frac{\\Pi}{2}-x\\right)\\right]^{5}}.dx}\\\\\n{I}&={\\int_{0}^{\\frac{\\Pi}{2}}\\frac{\\left[\\sin x\\right]^{5}}{\\left[\\cos x\\right]^{5}+ \\left[\\sin x\\right]^{5}}.dx}\\\\\n{I}&={\\int_{0}^{\\frac{\\Pi}{2}}\\frac{\\sin^{5}x}{\\sin^{5}x+\\cos^{5}x}.dx....\\left(2\\right)}\t\n\\end{align*}","ts":1602493284282,"cs":"FK1ZYG/z9glTEPWXhbBwUg==","size":{"width":312,"height":148}}

Add eq. 1 and 2:-

{"font":{"color":"#000000","family":"Arial","size":10},"type":"align*","code":"\\begin{align*}\n{2I}&={\\int_{0}^{\\frac{\\Pi}{2}}\\frac{\\cos^{5}x}{\\sin^{5}x+\\cos^{5}x}.dx+\\int_{0}^{\\frac{\\Pi}{2}}\\frac{\\sin^{5}x}{\\sin^{5}x+\\cos^{5}x}.dx}\\\\\n{2I}&={\\int_{0}^{\\frac{\\Pi}{2}}\\frac{\\cos^{5}x+\\sin^{5}x}{\\sin^{5}x+\\cos^{5}x}.dx}\\\\\n{2I}&={\\int_{0}^{\\frac{\\Pi}{2}}1.dx}\\\\\n{2I}&={\\left[x\\right]_{0}^{\\frac{\\Pi}{2}}}\\\\\n{2I}&={\\frac{\\Pi}{2}}\\\\\n{I\\,\\,}&={\\frac{\\Pi}{4}}\t\n\\end{align*}","id":"3-0-0-0-0-0-0-1-0-0-0-0-0-1-2-0-0-0","ts":1602493404573,"cs":"WKD5fGwG9/3siOacrEGkJg==","size":{"width":372,"height":236}}

{"id":"4-0-0-1-1-1-1-0-0","type":"$","font":{"size":10,"color":"#000000","family":"Arial"},"code":"$5.\\,\\int_{-5}^{5}\\left|x+2\\right|.dx$","ts":1602493482470,"cs":"1FVVZNBb3HaIm4PPRMFpuQ==","size":{"width":109,"height":21}}

Solun:- Let f(x) = |x + 2|

Integrate f(x):-

{"font":{"color":"#000000","size":10,"family":"Arial"},"type":"align*","id":"2-0-0-0-0-0-0-0-0-0-0-0-0-0-0-1-1-1-1-0-0","code":"\\begin{align*}\n{I}&={\\int_{-5}^{5}\\left|x+2\\right|.dx....\\left(1\\right)}\t\n\\end{align*}","ts":1602493522463,"cs":"SCLTyLNsi2yz7DVLYqe6wQ==","size":{"width":170,"height":40}}

By Property:-

{"type":"$","code":"$\\int_{a}^{b}f\\left(x\\right).dx=\\int_{a}^{c}f\\left(x\\right).dx+\\int_{c}^{b}f\\left(x\\right).dx$","font":{"size":10,"family":"Arial","color":"#000000"},"id":"1-2-2","ts":1602493604045,"cs":"i5/s8j6kPci7mHgn0Zayxg==","size":{"width":260,"height":21}}

{"code":"\\begin{align*}\n{I}&={\\int_{-5}^{-2}-\\left(x+2\\right).dx+\\int_{-2}^{5}\\left(x+2\\right).dx}\t\n\\end{align*}","type":"align*","id":"2-0-0-0-0-0-0-0-0-0-0-0-0-0-1-1-1-1-1-0-0","font":{"family":"Arial","size":10,"color":"#000000"},"ts":1602494153278,"cs":"5Pjeag/5kO9ibBM1wBQmgA==","size":{"width":264,"height":40}}

We know that:-

{"type":"$","font":{"color":"#000000","family":"Arial","size":10},"id":"7-0","code":"$\\int_{}^{}x^{n}.dx=\\frac{x^{n+1}}{n+1}+C$","ts":1602493752975,"cs":"ThIogt4GrHQEnxouMS0BCQ==","size":{"width":136,"height":21}}

{"font":{"family":"Arial","color":"#000000","size":10},"type":"align*","code":"\\begin{align*}\n{I}&={-\\left[\\frac{x^{2}}{2}+2x\\right]_{-5}^{-2}+\\left[\\frac{x^{2}}{2}+2x\\right]_{-2}^{5}}\\\\\n{I}&={-\\left[\\left(2-4\\right)-\\left(\\frac{25}{2}-10\\right)\\right]+\\left[\\left(\\frac{25}{2}+10\\right)-\\left(2-4\\right)\\right]}\\\\\n{I}&={-\\left[-2-\\left(\\frac{25}{2}-10\\right)\\right]+\\left[\\left(\\frac{25}{2}+10\\right)+2\\right]}\\\\\n{I}&={2+\\left(\\frac{25}{2}-10\\right)+\\left(\\frac{25}{2}+10\\right)+2}\\\\\n{I}&={4+25=29}\t\n\\end{align*}","id":"3-0-0-0-0-0-0-1-0-0-0-0-0-1-2-1-0-0","ts":1602494398110,"cs":"PtAQsDwpuIWrMPh7XZcqhw==","size":{"width":390,"height":192}}

{"id":"4-0-0-1-1-1-1-1-0","code":"$6.\\,\\int_{2}^{8}\\left|x-5\\right|.dx$","font":{"family":"Arial","color":"#000000","size":10},"type":"$","ts":1602494453376,"cs":"HxxhRb903Ilgh2zJy4+woA==","size":{"width":104,"height":20}}

Solun:- Let f(x) = |x - 5|

Integrate f(x):-

{"type":"align*","id":"2-0-0-0-0-0-0-0-0-0-0-0-0-0-0-1-1-1-1-1-0","code":"\\begin{align*}\n{I}&={\\int_{2}^{8}\\left|x-5\\right|.dx....\\left(1\\right)}\t\n\\end{align*}","font":{"family":"Arial","color":"#000000","size":10},"ts":1602494486169,"cs":"AubzpEnke6WhOWgOifaciw==","size":{"width":168,"height":38}}

By Property:-

{"type":"$","code":"$\\int_{a}^{b}f\\left(x\\right).dx=\\int_{a}^{c}f\\left(x\\right).dx+\\int_{c}^{b}f\\left(x\\right).dx$","font":{"size":10,"family":"Arial","color":"#000000"},"id":"1-2-2","ts":1602493604045,"cs":"i5/s8j6kPci7mHgn0Zayxg==","size":{"width":260,"height":21}}

{"code":"\\begin{align*}\n{I}&={\\int_{2}^{5}-\\left(x-5\\right).dx+\\int_{5}^{8}\\left(x-5\\right).dx}\t\n\\end{align*}","type":"align*","font":{"color":"#000000","family":"Arial","size":10},"id":"2-0-0-0-0-0-0-0-0-0-0-0-0-0-1-1-1-1-1-1","ts":1602766285696,"cs":"1XM2luOSVmZp6Jll/gvv7w==","size":{"width":253,"height":38}}

We know that:-

{"type":"$","font":{"color":"#000000","family":"Arial","size":10},"id":"7-1","code":"$\\int_{}^{}x^{n}.dx=\\frac{x^{n+1}}{n+1}+C$","ts":1602493752975,"cs":"JQypbQSr63m5h8/gx2nDfg==","size":{"width":136,"height":21}}

{"font":{"color":"#000000","size":10,"family":"Arial"},"code":"\\begin{align*}\n{I}&={-\\left[\\frac{x^{2}}{2}-5x\\right]_{2}^{5}+\\left[\\frac{x^{2}}{2}-5x\\right]_{5}^{8}}\\\\\n{I}&={-\\left[\\left(\\frac{25}{2}-25\\right)-\\left(2-10\\right)\\right]+\\left[\\left(32-40\\right)-\\left(\\frac{25}{2}-25\\right)\\right]}\\\\\n{I}&={-\\left[\\left(\\frac{25}{2}-25\\right)+8\\right]+\\left[-8-\\left(\\frac{25}{2}-25\\right)\\right]}\\\\\n{I}&={-\\left(\\frac{25}{2}-25\\right)-8-8-\\left(\\frac{25}{2}-25\\right)}\\\\\n{I}&={-16-2\\left(\\frac{25}{2}-25\\right)}\\\\\n{I}&={-16-25+50}\\\\\n{I}&={9}\t\n\\end{align*}","id":"3-0-0-0-0-0-0-1-0-0-0-0-0-1-2-1-1","type":"align*","ts":1602495631368,"cs":"cM6yMlpuiGnGxxXT5r2vzw==","size":{"width":413,"height":254}}

{"id":"4-0-0-1-1-1-1-1-1-0","type":"$","code":"$7.\\,\\int_{0}^{1}x\\left(1-x\\right)^{n}.dx$","font":{"family":"Arial","size":10,"color":"#000000"},"ts":1602495922427,"cs":"AdflOUMkZPiM74XbtdOcsA==","size":{"width":126,"height":21}}

Solun:- Let f(x) = x(1 - x)n

Integrate f(x):-

{"id":"2-0-0-0-0-0-0-0-0-0-0-0-0-0-0-1-1-1-1-1-1-0","type":"align*","font":{"color":"#000000","size":10,"family":"Arial"},"code":"\\begin{align*}\n{I}&={\\int_{0}^{1}x\\left(1-x\\right)^{n}.dx....\\left(1\\right)}\t\n\\end{align*}","ts":1602496100546,"cs":"RP3o3ie4zN6+wM4BGK/mdA==","size":{"width":192,"height":38}}

By Property:-

{"font":{"family":"Arial","color":"#000000","size":10},"type":"$","id":"1-2-1-1","code":"$\\int_{a}^{b}f\\left(x\\right).dx=\\int_{a}^{b}f\\left(a+b-x\\right).dx$","ts":1602403552089,"cs":"MOMCCBTEdVa0ebTt6VicFQ==","size":{"width":220,"height":21}}

{"font":{"size":10,"color":"#000000","family":"Arial"},"type":"align*","id":"2-0-0-0-0-0-0-0-0-0-0-0-0-0-1-1-1-1-0-1-0","code":"\\begin{align*}\n{I}&={\\int_{0}^{1}\\left(1-x\\right)\\left(1-\\left(1-x\\right)\\right)^{n}.dx}\\\\\n{I}&={\\int_{0}^{1}x^{n}\\left(1-x\\right).dx}\\\\\n{I}&={\\int_{0}^{1}\\left(x^{n}-x^{n+1}\\right).dx}\t\n\\end{align*}","ts":1602496551662,"cs":"vEQrvM6CA2IYqfL72rmC9A==","size":{"width":220,"height":126}}

We know that:-

{"type":"$","font":{"color":"#000000","family":"Arial","size":10},"id":"10","code":"$\\int_{}^{}x^{n}.dx=\\frac{x^{n+1}}{n+1}+C$","ts":1602493752975,"cs":"awXDTD168QNDUeWEhQgekQ==","size":{"width":136,"height":21}}

{"font":{"family":"Arial","color":"#000000","size":9.999999999999998},"id":"3-0-0-0-0-0-0-1-0-0-0-0-0-1-2-0-1-0","code":"\\begin{align*}\n{I}&={\\left[\\left(\\frac{x^{n+1}}{n+1}-\\frac{x^{n+2}}{n+2}\\right)\\right]_{0}^{1}}\\\\\n{I}&={\\left[\\left(\\frac{1}{n+1}-\\frac{1}{n+2}\\right)-\\left(0\\right)\\right]}\t\n\\end{align*}","type":"align*","ts":1602497154465,"cs":"JH/Xr//0o6No8ZI0Zvu//Q==","size":{"width":216,"height":85}}

{"font":{"color":"#000000","size":10,"family":"Arial"},"code":"\\begin{align*}\n{I}&={\\frac{n+2-n-1}{\\left(n+1\\right)\\left(n+2\\right)}}\\\\\n{I}&={\\frac{1}{\\left(n+1\\right)\\left(n+2\\right)}}\t\n\\end{align*}","type":"align*","id":"8-0","ts":1602497226990,"cs":"2MjHEYh4ArVw2uIUhBxapg==","size":{"width":133,"height":77}}

{"id":"4-0-0-1-1-1-1-1-1-1-0","type":"$","code":"$8.\\,\\int_{0}^{\\frac{\\Pi}{4}}\\log_{}\\left(1+\\tan x\\right).dx$","font":{"color":"#000000","family":"Arial","size":10},"ts":1602497304301,"cs":"/7YFOQU1sPR3h3O5lDJoZw==","size":{"width":164,"height":24}}

Solun:- Let f(x) = log (1 + tan x)

Integrate f(x):-

{"code":"\\begin{align*}\n{I}&={\\int_{0}^{\\frac{\\Pi}{4}}\\log_{}\\left(1+\\tan x\\right).dx....\\left(1\\right)}\t\n\\end{align*}","id":"2-0-0-0-0-0-0-0-0-0-0-0-0-0-0-1-1-1-1-1-1-1-0","type":"align*","font":{"family":"Arial","size":10,"color":"#000000"},"ts":1602497369074,"cs":"2OH0Y1byBzHFPotwmZFdug==","size":{"width":229,"height":40}}

By Property:-

{"font":{"family":"Arial","color":"#000000","size":10},"type":"$","id":"1-2-1-2","code":"$\\int_{a}^{b}f\\left(x\\right).dx=\\int_{a}^{b}f\\left(a+b-x\\right).dx$","ts":1602403552089,"cs":"d4n9hMBbOEs7Hrn7tkprNg==","size":{"width":220,"height":21}}

{"id":"2-0-0-0-0-0-0-0-0-0-0-0-0-0-1-1-1-1-0-1-1-0-0","font":{"color":"#000000","size":10,"family":"Arial"},"type":"align*","code":"\\begin{align*}\n{I}&={\\int_{0}^{\\frac{\\Pi}{4}}\\log_{}\\left(1+\\tan x\\right).dx}\\\\\n{I}&={\\int_{0}^{\\frac{\\Pi}{4}}\\log_{}\\left(1+\\tan\\left(\\frac{\\Pi}{4}-x\\right)\\right).dx}\t\n\\end{align*}","ts":1602497439686,"cs":"XGEGmG+9VXY36PFwLR6WSg==","size":{"width":253,"height":88}}

We know that:-

{"id":"9-0","type":"$","font":{"family":"Arial","color":"#000000","size":10},"code":"$\\tan\\left(A-B\\right)=\\frac{\\tan A-\\tan B}{1+\\tan A\\tan B}$","ts":1602497490011,"cs":"Em+GJcJB768bendkLOYrLw==","size":{"width":177,"height":20}}

{"id":"2-0-0-0-0-0-0-0-0-0-0-0-0-0-1-1-1-1-0-1-1-1-0-0","font":{"size":10,"color":"#000000","family":"Arial"},"code":"\\begin{align*}\n{I}&={\\int_{0}^{\\frac{\\Pi}{4}}\\log_{}\\left(1+\\left(\\frac{\\tan\\frac{\\Pi}{4}-\\tan x}{1+\\tan\\frac{\\Pi}{4}.\\tan x}\\right)\\right).dx}\\\\\n{I}&={\\int_{0}^{\\frac{\\Pi}{4}}\\log_{}\\left(1+\\left(\\frac{1-\\tan x}{1+\\tan x}\\right)\\right).dx}\\\\\n{I}&={\\int_{0}^{\\frac{\\Pi}{4}}\\log_{}\\left(\\frac{1+\\tan x+1-\\tan x}{1+\\tan x}\\right).dx}\\\\\n{I}&={\\int_{0}^{\\frac{\\Pi}{4}}\\log_{}\\left(\\frac{2}{1+\\tan x}\\right).dx}\\\\\n{I}&={\\int_{0}^{\\frac{\\Pi}{4}}\\left[\\log_{}\\left(2\\right)-\\log_{}\\left(1+\\tan x\\right)\\right].dx....\\left(2\\right)}\t\n\\end{align*}","type":"align*","ts":1602497983696,"cs":"llCphHYoAS445Oxtcwq3Sw==","size":{"width":301,"height":233}}

Add eq. 1 and 2:-

{"font":{"family":"Arial","size":10,"color":"#000000"},"type":"align*","code":"\\begin{align*}\n{2I}&={\\int_{0}^{\\frac{\\Pi}{4}}\\log_{}\\left(2\\right)-\\log_{}\\left(1+\\tan x\\right)+\\log_{}\\left(1+\\tan x\\right).dx}\\\\\n{2I}&={\\int_{0}^{\\frac{\\Pi}{4}}\\log_{}\\left(2\\right).dx}\\\\\n{2I}&={\\log_{}\\left(2\\right).\\int_{0}^{\\frac{\\Pi}{4}}1.dx}\t\n\\end{align*}","id":"2-0-0-0-0-0-0-0-0-0-0-0-0-0-1-1-1-1-0-1-1-1-1-0","ts":1602498182086,"cs":"TIjDfVtPAGj1lTm5d5MjyQ==","size":{"width":369,"height":133}}

{"id":"3-0-0-0-0-0-0-1-0-0-0-0-0-1-2-0-1-1-0","font":{"size":9.999999999999998,"family":"Arial","color":"#000000"},"code":"\\begin{align*}\n{I}&={\\frac{\\log_{}2}{2}\\left[x\\right]_{0}^{\\frac{\\Pi}{4}}}\\\\\n{I}&={\\frac{\\log_{}2}{2}\\times\\frac{\\Pi}{4}}\\\\\n{I}&={\\log_{}2\\times\\frac{\\Pi}{8}}\t\n\\end{align*}","type":"align*","ts":1602500560519,"cs":"ve6P0diDzDYVbnxRVscXyA==","size":{"width":105,"height":108}}

{"code":"$9.\\,\\int_{0}^{2}x{\\sqrt[]{2-x}}.dx$","type":"$","id":"4-0-0-1-1-1-1-1-1-1-1","font":{"family":"Arial","color":"#000000","size":10},"ts":1602500727320,"cs":"pKShBJ+t8pCm9PWjijLnjg==","size":{"width":120,"height":21}}

Solun:- Let f(x) = {"id":"12","code":"$x{\\sqrt[]{2-x}}$","font":{"size":10,"color":"#000000","family":"Arial"},"type":"$","ts":1602500769230,"cs":"86NQZ7ovwmzWjlTJFkjbbw==","size":{"width":58,"height":16}}

Integrate f(x):-

{"type":"align*","font":{"color":"#000000","size":10,"family":"Arial"},"id":"2-0-0-0-0-0-0-0-0-0-0-0-0-0-0-1-1-1-1-1-1-1-1","code":"\\begin{align*}\n{I}&={\\int_{0}^{2}x{\\sqrt[]{2-x}}.dx....\\left(1\\right)}\t\n\\end{align*}","ts":1602500795766,"cs":"YSl3lMGMZx53xMzEJ2sIaw==","size":{"width":185,"height":38}}

By Property:-

{"font":{"family":"Arial","color":"#000000","size":10},"type":"$","id":"1-2-1-3","code":"$\\int_{a}^{b}f\\left(x\\right).dx=\\int_{a}^{b}f\\left(a+b-x\\right).dx$","ts":1602403552089,"cs":"tyUHw3cH7C0drUcE4eGMRg==","size":{"width":220,"height":21}}

{"code":"\\begin{align*}\n{I}&={\\int_{0}^{2}\\left(2-x\\right){\\sqrt[]{2-\\left(2-x\\right)}}.dx}\\\\\n{I}&={\\int_{0}^{2}\\left(2-x\\right){\\sqrt[]{x}}.dx}\\\\\n{I}&={\\int_{0}^{2}\\left(2x^{\\frac{1}{2}}-x^{\\frac{3}{2}}\\right).dx}\t\n\\end{align*}","font":{"size":10,"family":"Arial","color":"#000000"},"id":"2-0-0-0-0-0-0-0-0-0-0-0-0-0-1-1-1-1-0-1-1-0-1","type":"align*","ts":1602501044943,"cs":"hEbxaHoc0iO3U0IYoVRk/A==","size":{"width":216,"height":126}}

We know that:-

{"type":"$","id":"9-1","font":{"color":"#000000","size":10,"family":"Arial"},"code":"$\\int_{}^{}x^{n}.dx=\\frac{x^{n+1}}{n+1}+C$","ts":1602501096313,"cs":"gMReBLlOpXj9iLUDz5xt7g==","size":{"width":136,"height":21}}

{"font":{"family":"Arial","color":"#000000","size":9.999999999999998},"type":"align*","code":"\\begin{align*}\n{I}&={\\left[\\frac{2x^{\\frac{3}{2}}}{\\frac{3}{2}}-\\frac{x^{\\frac{5}{2}}}{\\frac{5}{2}}\\right]_{0}^{2}}\\\\\n{I}&={\\left[\\frac{4x^{\\frac{3}{2}}}{3}-\\frac{2x^{\\frac{5}{2}}}{5}\\right]_{0}^{2}}\\\\\n{I}&={\\left[\\left(\\frac{8{\\sqrt[]{2}}}{3}-\\frac{8{\\sqrt[]{2}}}{5}\\right)-0\\right]}\\\\\n{I}&={\\left(\\frac{40{\\sqrt[]{2}}-24{\\sqrt[]{2}}}{15}\\right)}\\\\\n{I}&={\\frac{16{\\sqrt[]{2}}}{15}}\t\n\\end{align*}","id":"3-0-0-0-0-0-0-1-0-0-0-0-0-1-2-0-1-1-1-0","ts":1602501460260,"cs":"qeH/9qCZ0UcckePPQ5c38Q==","size":{"width":192,"height":253}}

{"type":"$","code":"$10.\\,\\int_{0}^{\\frac{\\Pi}{2}}\\left(2\\log_{}\\left(\\sin x\\right)-\\log_{}\\left(\\sin2x\\right)\\right).dx$","id":"4-0-0-1-1-1-0-1-0","font":{"family":"Arial","color":"#000000","size":10},"ts":1602501694980,"cs":"vk+iTRhoe6N1MT6HSRnF3A==","size":{"width":256,"height":24}}

Solun:- Let f(x) = 2log (sin x) - log (sin 2x)

Integrate f(x):-

{"type":"align*","code":"\\begin{align*}\n{I}&={\\int_{0}^{\\frac{\\Pi}{2}}\\left(2\\log_{}\\left(\\sin x\\right)-\\log_{}\\left(\\sin2x\\right)\\right).dx}\\\\\n{I}&={\\int_{0}^{\\frac{\\Pi}{2}}\\left(2\\log_{}\\left(\\sin x\\right)-\\log_{}\\left(2\\sin x.\\cos x\\right)\\right).dx....\\left(1\\right)}\t\n\\end{align*}","font":{"size":10,"color":"#000000","family":"Arial"},"id":"2-0-0-0-0-0-0-0-0-0-0-0-0-0-0-1-1-1-0-1-0-0","ts":1602501885613,"cs":"UqsluxoaqPEJU5A6+rAGTw==","size":{"width":356,"height":88}}

We know that:-

log (m.n) = log m + log n

log m - log n = log (m/n)

{"type":"align*","font":{"size":10,"family":"Arial","color":"#000000"},"id":"2-0-0-0-0-0-0-0-0-0-0-0-0-0-0-1-1-1-0-1-1","code":"\\begin{align*}\n{I}&={\\int_{0}^{\\frac{\\Pi}{2}}\\left(2\\log_{}\\left(\\sin x\\right)-\\log_{}\\left(2\\right)-\\log_{}\\left(\\sin x\\right)-\\log_{}\\left(\\cos x\\right)\\right).dx}\\\\\n{I}&={\\int_{0}^{\\frac{\\Pi}{2}}\\left(\\log_{}\\left(\\sin x\\right)-\\log_{}\\left(2\\right)-\\log_{}\\left(\\cos x\\right)\\right).dx}\\\\\n{I}&={\\int_{0}^{\\frac{\\Pi}{2}}\\left(\\log_{}\\left|\\frac{\\sin x}{\\cos x}\\right|-\\log_{}\\left(2\\right)\\right).dx}\\\\\n{I}&={\\int_{0}^{\\frac{\\Pi}{2}}\\left(\\log_{}\\left|\\tan x\\right|-\\log_{}\\left(2\\right)\\right).dx}\\\\\n{I}&={\\int_{0}^{\\frac{\\Pi}{2}}\\log_{}\\left|\\tan x\\right|.dx-\\int_{0}^{\\frac{\\Pi}{2}}\\log_{}2.dx....\\left(1\\right)}\t\n\\end{align*}","ts":1602502258844,"cs":"D2A/CZsA7ca9pLORPOLu4Q==","size":{"width":408,"height":226}}

By Property:-

{"font":{"family":"Arial","color":"#000000","size":10},"type":"$","id":"1-2-1-4-0","code":"$\\int_{a}^{b}f\\left(x\\right).dx=\\int_{a}^{b}f\\left(a+b-x\\right).dx$","ts":1602403552089,"cs":"LbHNA2mkV60wF43uVxx5JQ==","size":{"width":220,"height":21}}

{"font":{"color":"#000000","family":"Arial","size":10},"id":"2-0-0-0-0-0-0-0-0-0-0-0-0-0-1-1-1-1-0-0-1-0","type":"align*","code":"\\begin{align*}\n{I}&={\\int_{0}^{\\frac{\\Pi}{2}}\\log_{}\\left|\\tan \\left(\\frac{\\Pi}{2}-x\\right)\\right|.dx-\\int_{0}^{\\frac{\\Pi}{2}}\\log_{}2.dx}\\\\\n{I}&={\\int_{0}^{\\frac{\\Pi}{2}}\\log_{}\\left|\\cot x\\right|.dx-\\int_{0}^{\\frac{\\Pi}{2}}\\log_{}2.dx....\\left(2\\right)}\t\n\\end{align*}","ts":1602502379684,"cs":"imol0BTFh4xb29m1Rv/gGw==","size":{"width":316,"height":88}}

Add eq. 1 and 2:-

{"font":{"color":"#000000","size":10,"family":"Arial"},"id":"3-0-0-0-0-0-0-1-0-0-0-0-0-1-2-0-0-1-0","type":"align*","code":"\\begin{align*}\n{2I}&={\\int_{0}^{\\frac{\\Pi}{2}}\\log_{}\\left|\\cot x\\right|.dx-\\int_{0}^{\\frac{\\Pi}{2}}\\log_{}2.dx+\\int_{0}^{\\frac{\\Pi}{2}}\\log_{}\\left|\\tan x\\right|.dx-\\int_{0}^{\\frac{\\Pi}{2}}\\log_{}2.dx}\\\\\n{2I}&={\\int_{0}^{\\frac{\\Pi}{2}}\\left(\\log_{}\\left|\\cot x\\right|+\\log_{}\\left|\\tan x\\right|\\right).dx-2\\int_{0}^{\\frac{\\Pi}{2}}\\log_{}2.dx}\\\\\n{2I}&={\\int_{0}^{\\frac{\\Pi}{2}}\\log_{}\\left|\\cot x.\\tan x\\right|.dx-2\\int_{0}^{\\frac{\\Pi}{2}}\\log_{}2.dx}\\\\\n{2I}&={\\int_{0}^{\\frac{\\Pi}{2}}\\log_{}\\left|1\\right|.dx-2\\int_{0}^{\\frac{\\Pi}{2}}\\log_{}2.dx}\\\\\n{2I}&={0-2\\int_{0}^{\\frac{\\Pi}{2}}\\log_{}2.dx}\\\\\n{2I\\,\\,}&={-2\\log_{}2.\\int_{0}^{\\frac{\\Pi}{2}}1.dx}\t\n\\end{align*}","ts":1602502690989,"cs":"qxWhF7qEddEQruly8jCEsg==","size":{"width":508,"height":272}}

{"font":{"family":"Arial","size":9.999999999999998,"color":"#000000"},"id":"3-0-0-0-0-0-0-1-0-0-0-0-0-1-2-0-1-1-1-1-0","type":"align*","code":"\\begin{align*}\n{I}&={-\\log_{}2.\\left[x\\right]_{0}^{\\frac{\\Pi}{2}}}\\\\\n{I}&={-\\frac{\\Pi}{2}\\log_{}2}\\\\\n{I}&={\\frac{\\Pi}{2}\\log_{}\\frac{1}{2}}\t\n\\end{align*}","ts":1602502898793,"cs":"02MKrSIDTGxmgFtZHiAwvw==","size":{"width":109,"height":97}}

{"code":"$11.\\,\\int_{\\frac{-\\Pi}{2}}^{\\frac{\\Pi}{2}}\\sin^{2}x.dx$","font":{"color":"#000000","family":"Arial","size":10},"id":"4-0-0-1-1-1-0-1-1-0","type":"$","ts":1602503065486,"cs":"uKpYlcnn3ezAlfkhbqHngA==","size":{"width":114,"height":29}}

Solun:- Let f(x) = sin2x

Integrate f(x):-

{"type":"align*","id":"2-0-0-0-0-0-0-0-0-0-0-0-0-0-0-1-1-1-0-1-0-1-0","code":"\\begin{align*}\n{I}&={\\int_{\\frac{-\\Pi}{2}}^{\\frac{\\Pi}{2}}\\sin^{2}x.dx}\t\n\\end{align*}","font":{"size":10,"family":"Arial","color":"#000000"},"ts":1602503271826,"cs":"VraUn0Uxp+dr1K+D8WfP0g==","size":{"width":120,"height":44}}

By Property:-

{"font":{"color":"#000000","family":"Arial","size":10},"id":"1-2-1-5-0","code":"\\begin{align*}\n{\\int_{-a}^{a}f\\left(x\\right).dx}&={2\\int_{0}^{a}f\\left(x\\right).dx,if\\,f\\left(-x\\right)=f\\left(x\\right)}\\\\\n{\\int_{-a}^{a}f\\left(x\\right).dx}&={0,if\\,f\\left(-x\\right)=-f\\left(x\\right)}\t\n\\end{align*}","type":"align*","ts":1602503490672,"cs":"tgOYrM8IY0JSDoJMmt349A==","size":{"width":309,"height":78}}

f(-x) = [sin (-x)]2 = [-sin x]2

f(-x) = sin2x = f(x)

{"type":"align*","code":"\\begin{align*}\n{I}&={2\\int_{0}^{\\frac{\\Pi}{2}}\\sin^{2}x.dx...\\left(1\\right)}\t\n\\end{align*}","font":{"color":"#000000","family":"Arial","size":10},"id":"2-0-0-0-0-0-0-0-0-0-0-0-0-0-1-1-1-1-0-0-1-1-0-0-0","ts":1602503740043,"cs":"iyDTkgByqfTFwxapSfa+Lw==","size":{"width":172,"height":40}}

By property:-

{"font":{"family":"Arial","color":"#000000","size":10},"type":"$","id":"1-2-1-4-1","code":"$\\int_{a}^{b}f\\left(x\\right).dx=\\int_{a}^{b}f\\left(a+b-x\\right).dx$","ts":1602403552089,"cs":"1rvls64O5BA4di5QAzBp1Q==","size":{"width":220,"height":21}}

{"id":"2-0-0-0-0-0-0-0-0-0-0-0-0-0-1-1-1-1-0-0-1-1-1-0-0","type":"align*","font":{"size":10,"color":"#000000","family":"Arial"},"code":"\\begin{align*}\n{I}&={2\\int_{0}^{\\frac{\\Pi}{2}}\\left[\\sin \\left(\\frac{\\Pi}{2}-x\\right)\\right]^{2}.dx}\\\\\n{I}&={2\\int_{0}^{\\frac{\\Pi}{2}}\\left[\\cos x\\right]^{2}.dx}\\\\\n{I}&={2\\int_{0}^{\\frac{\\Pi}{2}}\\cos ^{2}x.dx....\\left(2\\right)}\t\n\\end{align*}","ts":1602503945254,"cs":"XXjohLvHM7N2ZWkc/s171w==","size":{"width":209,"height":134}}

Add eq. 1 and 2:-

{"font":{"color":"#000000","family":"Arial","size":10},"type":"align*","id":"3-0-0-0-0-0-0-1-0-0-0-0-0-1-2-0-0-1-1-0-0","code":"\\begin{align*}\n{2I}&={2\\int_{0}^{\\frac{\\Pi}{2}}\\cos^{2}x+\\sin^{2}x.dx}\\\\\n{2I}&={2\\int_{0}^{\\frac{\\Pi}{2}}1.dx}\t\n\\end{align*}","ts":1602504808065,"cs":"yGMgnjZyfXJbg4P8BCKuTw==","size":{"width":197,"height":88}}

{"code":"\\begin{align*}\n{I}&={\\left[x\\right]_{0}^{\\frac{\\Pi}{2}}}\\\\\n{I}&={\\frac{\\Pi}{2}}\t\n\\end{align*}","id":"3-0-0-0-0-0-0-1-0-0-0-0-0-1-2-0-1-1-1-1-1-0","font":{"family":"Arial","size":9.999999999999998,"color":"#000000"},"type":"align*","ts":1602768581803,"cs":"EecLyfIPJTQ2ZLfpxf3ERA==","size":{"width":58,"height":60}}

{"id":"4-0-0-1-1-1-0-1-1-1-0-0","type":"$","code":"$12.\\,\\int_{0}^{\\Pi}\\frac{x}{1+\\sin x}.dx$","font":{"family":"Arial","size":10,"color":"#000000"},"ts":1602505141820,"cs":"NBI98w4DsAYNQ8fIOOI3aQ==","size":{"width":116,"height":21}}

Solun:- Let f(x) = {"code":"$\\frac{x}{1+\\sin x}$","id":"13","type":"$","font":{"color":"#000000","size":10,"family":"Arial"},"ts":1602505165578,"cs":"HI7ekR1OvhvgYlvOOhG8qQ==","size":{"width":42,"height":17}}

Integrate f(x):-

{"font":{"color":"#000000","size":10,"family":"Arial"},"id":"2-0-0-0-0-0-0-0-0-0-0-0-0-0-0-1-1-1-0-1-0-1-1-0-0","code":"\\begin{align*}\n{I}&={\\int_{0}^{\\Pi}\\frac{x}{1+\\sin x}.dx....\\left(1\\right)}\t\n\\end{align*}","type":"align*","ts":1602508534202,"cs":"qwnJ2qkRW1Uaw0gMMBwf0A==","size":{"width":194,"height":38}}

By property:-

{"font":{"family":"Arial","color":"#000000","size":10},"type":"$","id":"1-2-1-4-2","code":"$\\int_{a}^{b}f\\left(x\\right).dx=\\int_{a}^{b}f\\left(a+b-x\\right).dx$","ts":1602403552089,"cs":"wiHTGT4himonjqRjtTJvpQ==","size":{"width":220,"height":21}}

{"id":"2-0-0-0-0-0-0-0-0-0-0-0-0-0-1-1-1-1-0-0-1-1-1-1","type":"align*","code":"\\begin{align*}\n{I\\,\\,}&={\\int_{0}^{\\Pi}\\frac{\\Pi-x}{1+\\sin\\left(\\Pi-x\\right)}.dx}\\\\\n{I\\,\\,}&={\\int_{0}^{\\Pi}\\frac{\\Pi}{1+\\sin x}.dx-\\int_{0}^{\\Pi}\\frac{x}{1+\\sin x}.dx}\\\\\n{I\\,\\,}&={\\int_{0}^{\\Pi}\\frac{\\Pi}{1+\\sin x}.dx-I\\,\\,\\left(From\\,eq.\\,1\\right)}\\\\\n{2I}&={\\Pi\\int_{0}^{\\Pi}\\frac{1}{1+\\sin x}.dx}\\\\\n{I\\,\\,}&={\\frac{\\Pi}{2}\\int_{0}^{\\Pi}\\frac{1}{1+\\sin x}\\times\\frac{1-\\sin x}{1-\\sin x}.dx}\\\\\n{I\\,\\,}&={\\frac{\\Pi}{2}\\int_{0}^{\\Pi}\\frac{1-\\sin x}{1-\\sin ^{2}x}.dx}\\\\\n{I\\,\\,}&={\\frac{\\Pi}{2}\\int_{0}^{\\Pi}\\frac{1-\\sin x}{ \\cos^{2}x}.dx}\\\\\n{I\\,\\,}&={\\frac{\\Pi}{2}\\left[\\int_{0}^{\\Pi}\\frac{1}{ \\cos^{2}x}.dx-\\int_{0}^{\\Pi}\\frac{\\sin x}{ \\cos^{2}x}.dx\\right]}\\\\\n{I\\,\\,}&={\\frac{\\Pi}{2}\\left[\\int_{0}^{\\Pi}\\sec^{2}x.dx-\\int_{0}^{\\Pi}\\sec x.\\tan x.dx\\right]}\t\n\\end{align*}","font":{"family":"Arial","color":"#000000","size":10},"ts":1602508510332,"cs":"r5wBaRJ/aB4BFKh6+/svTA==","size":{"width":308,"height":393}}

{"code":"\\begin{align*}\n{I}&={\\frac{\\Pi}{2}\\left[\\tan x-\\sec x\\right]_{0}^{\\Pi}}\\\\\n{I}&={\\frac{\\Pi}{2}\\left[\\left(0-\\left(-1\\right)\\right)-\\left(0-1\\right)\\right]}\\\\\n{I}&={\\frac{\\Pi}{2}\\times2=\\Pi}\t\n\\end{align*}","font":{"color":"#000000","family":"Arial","size":9.999999999999998},"type":"align*","id":"3-0-0-0-0-0-0-1-0-0-0-0-0-1-2-0-1-1-1-1-1-1","ts":1602508786973,"cs":"o8l19YKXL0b0CNHK28tWFQ==","size":{"width":196,"height":106}}

{"id":"4-0-0-1-1-1-0-1-1-1-1","code":"$13.\\,\\int_{\\frac{-\\Pi}{2}}^{\\frac{\\Pi}{2}}\\sin^{7}x.dx$","type":"$","font":{"color":"#000000","size":10,"family":"Arial"},"ts":1602583354163,"cs":"UbQM3XTUJZwqYbxP4A47Xg==","size":{"width":114,"height":29}}

Solun:- Let f(x) = sin7x

Integrate f(x):-

{"font":{"family":"Arial","color":"#000000","size":10},"type":"align*","id":"2-0-0-0-0-0-0-0-0-0-0-0-0-0-0-1-1-1-0-1-0-1-1-1-0-0","code":"\\begin{align*}\n{I}&={\\int_{\\frac{-\\Pi}{2}}^{\\frac{\\Pi}{2}}\\sin^{7}x.dx....\\left(1\\right)}\t\n\\end{align*}","ts":1602583441215,"cs":"QbzYL9zJKHxdpIIo2LeJag==","size":{"width":169,"height":44}}

By Property:-

{"font":{"color":"#000000","family":"Arial","size":10},"id":"1-2-1-5-1","code":"\\begin{align*}\n{\\int_{-a}^{a}f\\left(x\\right).dx}&={2\\int_{0}^{a}f\\left(x\\right).dx,if\\,f\\left(-x\\right)=f\\left(x\\right)}\\\\\n{\\int_{-a}^{a}f\\left(x\\right).dx}&={0,if\\,f\\left(-x\\right)=-f\\left(x\\right)}\t\n\\end{align*}","type":"align*","ts":1602503490672,"cs":"ju0dUPVndQfGk7GL29ALLg==","size":{"width":309,"height":78}}

f(-x) = [sin (-x)]7 = [-sin x]7

f(-x) = -sin7x = - f(x)

{"code":"\\begin{align*}\n{I}&={\\int_{\\frac{-\\Pi}{2}}^{\\frac{\\Pi}{2}}\\sin^{7}x.dx=0}\t\n\\end{align*}","type":"align*","font":{"family":"Arial","color":"#000000","size":10},"id":"2-0-0-0-0-0-0-0-0-0-0-0-0-0-0-1-1-1-0-1-0-1-1-1-1","ts":1602583667525,"cs":"S4+RmbbJKBBAjs5MCxv/Ng==","size":{"width":149,"height":44}}

{"type":"$","code":"$14.\\,\\int_{0}^{2\\Pi}\\cos^{5}x.dx$","id":"4-0-0-1-1-1-0-1-1-1-0-1-0","font":{"size":10,"family":"Arial","color":"#000000"},"ts":1602584141670,"cs":"S1bSkVS/EWtRWOuDrH+hog==","size":{"width":116,"height":21}}

Solun:- Let f(x) = cos5x

Integrate f(x):-

{"code":"\\begin{align*}\n{I}&={\\int_{0}^{2\\Pi}\\cos^{5}x.dx}\t\n\\end{align*}","id":"2-0-0-0-0-0-0-0-0-0-0-0-0-0-0-1-1-1-0-1-0-1-1-0-1-0","font":{"family":"Arial","size":10,"color":"#000000"},"type":"align*","ts":1602585041176,"cs":"MmxQ+jAZsYBaUfrnA7RxSQ==","size":{"width":125,"height":38}}

By Property:-

{"id":"1-2-1-5-2","font":{"color":"#000000","size":10,"family":"Arial"},"type":"align*","code":"\\begin{align*}\n{\\int_{0}^{2a}f\\left(x\\right).dx}&={2\\int_{0}^{a}f\\left(x\\right).dx,if\\,f\\left(2a-x\\right)=f\\left(x\\right)}\\\\\n{}&={0,if\\,f\\left(2a-x\\right)=-f\\left(x\\right)}\t\n\\end{align*}","ts":1602584511302,"cs":"trbLJOb2EThzHj6YKjCulQ==","size":{"width":337,"height":60}}

f(2a-x) = [cos (2Ï€-x)]5 = [cos x]5

f(2a-x) = cos5x = f(x)

{"id":"2-0-0-0-0-0-0-0-0-0-0-0-0-0-1-1-1-1-0-0-1-1-0-0-1","code":"\\begin{align*}\n{I}&={2\\int_{0}^{\\Pi}\\cos^{5}x.dx....\\left(1\\right)}\t\n\\end{align*}","type":"align*","font":{"size":10,"family":"Arial","color":"#000000"},"ts":1602585021092,"cs":"wLEhtxG/EVUTGqtzK/woRQ==","size":{"width":180,"height":38}}

By property:-

{"font":{"family":"Arial","color":"#000000","size":10},"type":"$","id":"1-2-1-4-3","code":"$\\int_{a}^{b}f\\left(x\\right).dx=\\int_{a}^{b}f\\left(a+b-x\\right).dx$","ts":1602403552089,"cs":"jSt/hya3wZeLoYqmRZin4g==","size":{"width":220,"height":21}}

{"code":"\\begin{align*}\n{I}&={2\\int_{0}^{\\Pi}\\left[\\cos\\left(\\Pi-x\\right)\\right]^{5}.dx}\\\\\n{I}&={2\\int_{0}^{\\Pi}\\left[-\\cos x\\right]^{5}.dx}\\\\\n{I}&={-2\\int_{0}^{\\Pi}\\cos ^{5}x.dx....\\left(2\\right)}\t\n\\end{align*}","font":{"color":"#000000","family":"Arial","size":10},"id":"2-0-0-0-0-0-0-0-0-0-0-0-0-0-1-1-1-1-0-0-1-1-1-0-1-0","type":"align*","ts":1602853407370,"cs":"7SAxN8eSG9jo3GuZwkHXLQ==","size":{"width":192,"height":126}}

Add eq. 1 and 2:-

{"type":"align*","code":"\\begin{align*}\n{2I}&={2\\int_{0}^{\\Pi}\\cos^{5}x.dx-2\\int_{0}^{\\Pi}\\cos^{5}x.dx}\\\\\n{2I}&={2\\left[\\int_{0}^{\\Pi}\\cos^{5}x.dx-\\int_{0}^{\\Pi}\\cos^{5}x.dx\\right]}\\\\\n{2I}&={0}\\\\\n{I\\,\\,}&={0}\t\n\\end{align*}","font":{"size":10,"family":"Arial","color":"#000000"},"id":"3-0-0-0-0-0-0-1-0-0-0-0-0-1-2-0-0-1-1-0-1-0","ts":1602585240406,"cs":"g5Vu94PblDmFAep5+c7xzA==","size":{"width":264,"height":124}}

{"code":"$15.\\,\\int_{0}^{\\frac{\\Pi}{2}}\\frac{\\sin x-\\cos x}{1+\\sin x.\\cos x}.dx$","type":"$","id":"4-0-0-1-1-1-0-1-1-1-0-1-1-0","font":{"color":"#000000","family":"Arial","size":10},"ts":1602585347805,"cs":"2//s+m871bZUy5Ya8qzS2g==","size":{"width":144,"height":25}}

Solun:- Let f(x) = {"code":"$\\frac{\\sin x-\\cos x}{1+\\sin x.\\cos x}$","id":"14","type":"$","font":{"color":"#000000","family":"Arial","size":10},"ts":1602585374201,"cs":"JKxjEdtjP3Eadl8uLnwT8w==","size":{"width":69,"height":20}}

Integrate f(x):-

{"id":"2-0-0-0-0-0-0-0-0-0-0-0-0-0-0-1-1-1-0-1-0-1-1-0-1-1-0","font":{"size":10,"family":"Arial","color":"#000000"},"type":"align*","code":"\\begin{align*}\n{I}&={\\int_{0}^{\\frac{\\Pi}{2}}\\frac{\\sin x-\\cos x}{1+\\sin x.\\cos x}.dx....\\left(1\\right)}\t\n\\end{align*}","ts":1602585514684,"cs":"KfJLau5f3hpfmtclswRTzA==","size":{"width":237,"height":40}}

By property:-

{"font":{"family":"Arial","color":"#000000","size":10},"type":"$","id":"1-2-1-4-4","code":"$\\int_{a}^{b}f\\left(x\\right).dx=\\int_{a}^{b}f\\left(a+b-x\\right).dx$","ts":1602403552089,"cs":"qmAjk++9vUqnu+Za9uRH4w==","size":{"width":220,"height":21}}

{"code":"\\begin{align*}\n{I}&={\\int_{0}^{\\frac{\\Pi}{2}}\\frac{\\sin \\left(\\frac{\\Pi}{2}-x\\right)-\\cos \\left(\\frac{\\Pi}{2}-x\\right)}{1+\\sin \\left(\\frac{\\Pi}{2}-x\\right).\\cos \\left(\\frac{\\Pi}{2}-x\\right)}.dx}\\\\\n{I}&={\\int_{0}^{\\frac{\\Pi}{2}}\\frac{\\cos x-\\sin x}{1+\\cos x.\\sin x}.dx....\\left(2\\right)}\t\n\\end{align*}","font":{"family":"Arial","size":10,"color":"#000000"},"id":"2-0-0-0-0-0-0-0-0-0-0-0-0-0-1-1-1-1-0-0-1-1-1-0-1-1-0","type":"align*","ts":1602585632441,"cs":"PgM9mQED+2+j02mnbjwEtQ==","size":{"width":288,"height":92}}

Add eq. 1 and 2:-

{"code":"\\begin{align*}\n{2I}&={\\int_{0}^{\\frac{\\Pi}{2}}\\frac{\\sin x-\\cos x}{1+\\cos x.\\sin x}.dx+\\int_{0}^{\\frac{\\Pi}{2}}\\frac{\\cos x-\\sin x}{1+\\cos x.\\sin x}.dx}\\\\\n{2I}&={\\int_{0}^{\\frac{\\Pi}{2}}\\frac{\\sin x-\\cos x+\\cos x-\\sin x}{1+\\cos x.\\sin x}.dx}\\\\\n{2I}&={0}\\\\\n{I\\,\\,}&={0}\t\n\\end{align*}","font":{"family":"Arial","size":10,"color":"#000000"},"type":"align*","id":"3-0-0-0-0-0-0-1-0-0-0-0-0-1-2-0-0-1-1-0-1-1-0","ts":1602585797230,"cs":"lVzP9E+JkfW+/zXTyamNVw==","size":{"width":376,"height":128}}

{"font":{"family":"Arial","color":"#000000","size":10},"code":"$16.\\,\\int_{0}^{\\Pi}\\log_{}\\left(1+\\cos x\\right).dx$","type":"$","id":"4-0-0-1-1-1-0-1-1-1-0-1-1-1","ts":1602585870108,"cs":"fSZTeD/2zdHPYIANflOdFA==","size":{"width":168,"height":21}}

Solun:- Let f(x) = log (1 + cos x)

Integrate f(x):-

{"type":"align*","id":"2-0-0-0-0-0-0-0-0-0-0-0-0-0-0-1-1-1-0-1-0-1-1-0-1-1-1","font":{"family":"Arial","color":"#000000","size":10},"code":"\\begin{align*}\n{I}&={\\int_{0}^{\\Pi}\\log_{}\\left(1+\\cos x\\right).dx....\\left(1\\right)}\t\n\\end{align*}","ts":1602585925891,"cs":"6k3PHr9bc0L7/8FwoY6qBQ==","size":{"width":225,"height":38}}

By property:-

{"font":{"family":"Arial","color":"#000000","size":10},"type":"$","id":"1-2-1-4-5","code":"$\\int_{a}^{b}f\\left(x\\right).dx=\\int_{a}^{b}f\\left(a+b-x\\right).dx$","ts":1602403552089,"cs":"Xa+gL3Ch7fmMxYKLii0Jjg==","size":{"width":220,"height":21}}

{"code":"\\begin{align*}\n{I}&={\\int_{0}^{\\Pi}\\log_{}\\left(1+\\cos \\left(\\Pi -x\\right)\\right).dx}\\\\\n{I}&={\\int_{0}^{\\Pi}\\log_{}\\left(1-\\cos x\\right).dx....\\left(2\\right)}\t\n\\end{align*}","id":"2-0-0-0-0-0-0-0-0-0-0-0-0-0-1-1-1-1-0-0-1-1-1-0-1-1-1","type":"align*","font":{"size":10,"color":"#000000","family":"Arial"},"ts":1602586069461,"cs":"rQf2ljIMcnONhVIUtkKJaw==","size":{"width":225,"height":82}}

Add eq. 1 and 2:-

{"id":"3-0-0-0-0-0-0-1-0-0-0-0-0-1-2-0-0-1-1-0-1-1-1-0-0","font":{"family":"Arial","size":10,"color":"#000000"},"code":"\\begin{align*}\n{2I}&={\\int_{0}^{\\Pi}\\log_{}\\left(1+\\cos x\\right).dx+\\int_{0}^{\\Pi}\\log_{}\\left(1-\\cos x\\right).dx}\\\\\n{2I}&={\\int_{0}^{\\Pi}\\log_{}\\left[\\left(1+\\cos x\\right)\\left(1-\\cos x\\right)\\right].dx}\\\\\n{2I}&={\\int_{0}^{\\Pi}\\log_{}\\left(1-\\cos ^{2}x\\right).dx}\\\\\n{2I}&={\\int_{0}^{\\Pi}\\log_{}\\left( \\sin^{2}x\\right).dx}\\\\\n{2I}&={2\\int_{0}^{\\Pi}\\log_{}\\left( \\sin x\\right).dx}\\\\\n{I\\,\\,}&={\\int_{0}^{\\Pi}\\log_{}\\left( \\sin x\\right).dx}\\\\\n{I\\,\\,}&={2\\int_{0}^{\\frac{\\Pi}{2}}\\log_{}\\left( \\sin x\\right).dx...\\left(3\\right)}\t\n\\end{align*}","type":"align*","ts":1602586743926,"cs":"kFshdxHLfOI5ugl2J92U6g==","size":{"width":352,"height":305}}

By property:-

{"font":{"family":"Arial","color":"#000000","size":10},"type":"$","id":"1-2-1-4-6","code":"$\\int_{a}^{b}f\\left(x\\right).dx=\\int_{a}^{b}f\\left(a+b-x\\right).dx$","ts":1602403552089,"cs":"iqYnXwbllRi40VtN0SbtPw==","size":{"width":220,"height":21}}

{"type":"align*","font":{"family":"Arial","size":10,"color":"#000000"},"code":"\\begin{align*}\n{I\\,\\,}&={2\\int_{0}^{\\frac{\\Pi}{2}}\\log_{}\\left( \\sin \\left(\\frac{\\Pi}{2}-x\\right)\\right).dx}\\\\\n{I\\,\\,}&={2\\int_{0}^{\\frac{\\Pi}{2}}\\log_{}\\left( \\cos x\\right).dx....\\left(4\\right)}\t\n\\end{align*}","id":"3-0-0-0-0-0-0-1-0-0-0-0-0-1-2-0-0-1-1-0-1-1-1-1","ts":1602587004919,"cs":"gyruU3HaP6ITlGPwQKHBqA==","size":{"width":238,"height":88}}

Add eq. 3 and 4:-

{"type":"align*","id":"3-0-0-0-0-0-0-1-0-0-0-0-0-1-2-0-0-1-1-0-1-1-1-0-1-0","font":{"size":10,"family":"Arial","color":"#000000"},"code":"\\begin{align*}\n{2I}&={2\\int_{0}^{\\frac{\\Pi}{2}}\\log_{}\\left(\\sin x\\right).dx+2\\int_{0}^{\\frac{\\Pi}{2}}\\log_{}\\left(\\cos x\\right).dx}\\\\\n{2I}&={2\\left[\\int_{0}^{\\frac{\\Pi}{2}}\\log_{}\\left(\\sin x\\right).dx+\\int_{0}^{\\frac{\\Pi}{2}}\\log_{}\\left(\\cos x\\right).dx\\right]}\\\\\n{2I}&={2\\int_{0}^{\\frac{\\Pi}{2}}\\log_{}\\left(\\sin x.\\cos x\\right).dx}\\\\\n{2I}&={2\\int_{0}^{\\frac{\\Pi}{2}}\\log_{}\\left(\\frac{2\\sin x.\\cos x}{2}\\right).dx}\\\\\n{2I}&={2\\int_{0}^{\\frac{\\Pi}{2}}\\log_{}\\left(\\frac{\\sin 2x}{2}\\right).dx}\\\\\n{2I}&={2\\left[\\int_{0}^{\\frac{\\Pi}{2}}\\log_{}\\left(\\sin 2x\\right).dx-\\int_{0}^{\\frac{\\Pi}{2}}\\log_{}\\left(2\\right).dx\\right]}\t\n\\end{align*}","ts":1602587549785,"cs":"eYMaNZznrBaxzDxHw7XLjg==","size":{"width":328,"height":285}}

Let 2x = t........(A)

Differentiate w.r.t. to t:-

2.dx = dt

dx = (1/2).dt

Limits Change:-

Put x = 0 in eq. A:-

t = 0

Put x = π/2 in eq. A:-

t = π

{"id":"3-0-0-0-0-0-0-1-0-0-0-0-0-1-2-0-0-1-1-0-1-1-1-0-1-1-0","code":"\\begin{align*}\n{2I}&={2\\left[\\frac{1}{2}\\int_{0}^{\\Pi}\\log_{}\\left(\\sin t\\right).dt-\\int_{0}^{\\frac{\\Pi}{2}}\\log_{}\\left(2\\right).dx\\right]}\\\\\n{I\\,\\,}&={\\frac{1}{2}\\int_{0}^{\\Pi}\\log_{}\\left(\\sin t\\right).dt-\\int_{0}^{\\frac{\\Pi}{2}}\\log_{}\\left(2\\right).dx}\t\n\\end{align*}","font":{"color":"#000000","family":"Arial","size":10},"type":"align*","ts":1602587986016,"cs":"W+a8QBd8I+mRaKx/bbDOXg==","size":{"width":312,"height":93}}

By Property:-

{"type":"$","font":{"color":"#000000","family":"Arial","size":10},"code":"$\\int_{a}^{b}f\\left(x\\right).dx=\\int_{a}^{b}f\\left(t\\right).dt$","id":"1-2-1-4-7","ts":1602588078614,"cs":"6I9MJPFw4V99xfgnwZtc2A==","size":{"width":160,"height":21}}

{"code":"\\begin{align*}\n{I}&={\\frac{1}{2}\\int_{0}^{\\Pi}\\log_{}\\left(\\sin x\\right).dx-\\int_{0}^{\\frac{\\Pi}{2}}\\log_{}\\left(2\\right).dx}\\\\\n{I}&={\\frac{2}{2}\\int_{0}^{\\frac{\\Pi}{2}}\\log_{}\\left(\\sin x\\right).dx-\\int_{0}^{\\frac{\\Pi}{2}}\\log_{}\\left(2\\right).dx}\\\\\n{I}&={\\int_{0}^{\\frac{\\Pi}{2}}\\log_{}\\left(\\sin x\\right).dx-\\int_{0}^{\\frac{\\Pi}{2}}\\log_{}\\left(2\\right).dx}\t\n\\end{align*}","font":{"size":10,"color":"#000000","family":"Arial"},"type":"align*","id":"3-0-0-0-0-0-0-1-0-0-0-0-0-1-2-0-0-1-1-0-1-1-1-0-1-1-1-0","ts":1602588256007,"cs":"JFNSQUDa7yYe5eAiWqdcmg==","size":{"width":285,"height":133}}

By eq. 3:-

{"code":"\\begin{align*}\n{I}&={\\frac{I}{2}-\\int_{0}^{\\frac{\\Pi}{2}}\\log_{}\\left(2\\right).dx}\\\\\n{\\frac{I}{2}}&={-\\int_{0}^{\\frac{\\Pi}{2}}\\log_{}\\left(2\\right).dx}\\\\\n{I}&={-2\\log_{}2\\int_{0}^{\\frac{\\Pi}{2}}1.dx}\\\\\n{I}&={-2\\log_{}2\\left[x\\right]_{0}^{\\frac{\\Pi}{2}}}\\\\\n{I}&={-2\\log_{}2\\times\\frac{\\Pi}{2}}\\\\\n{I}&={-\\Pi\\log_{}2}\t\n\\end{align*}","id":"3-0-0-0-0-0-0-1-0-0-0-0-0-1-2-0-0-1-1-0-1-1-1-0-1-1-1-1","font":{"size":10,"color":"#000000","family":"Arial"},"type":"align*","ts":1602588556223,"cs":"gQ8OOztgpHA3CykEyMkm5Q==","size":{"width":168,"height":220}}

{"font":{"color":"#000000","size":11,"family":"Arial"},"id":"4-0-0-1-0-1","type":"$","code":"$17.\\,\\int_{0}^{a}\\frac{{\\sqrt[]{x}}}{{\\sqrt[]{x}}+{\\sqrt[]{a-x}}}.dx$","ts":1602588732269,"cs":"cAVSJW9g20rL4KYLryU2/Q==","size":{"width":150,"height":30}}

Solun:- Let f(x) = {"code":"$\\frac{{\\sqrt[]{x}}}{{\\sqrt[]{x}}+{\\sqrt[]{a-x}}}$","id":"16-0","type":"$","font":{"color":"#000000","size":10,"family":"Arial"},"ts":1602588752858,"cs":"330irwnTVr6Eg+pLBYNv6A==","size":{"width":61,"height":26}}

Integrate f(x):-

{"code":"\\begin{align*}\n{I}&={\\int_{0}^{a}\\frac{{\\sqrt[]{x}}}{{\\sqrt[]{x}}+{\\sqrt[]{a-x}}}.dx....\\left(1\\right)}\t\n\\end{align*}","type":"align*","id":"2-0-0-0-0-0-0-0-0-0-0-0-0-0-0-1-0-1-0","font":{"family":"Arial","size":10,"color":"#000000"},"ts":1602588773599,"cs":"7LZOmDoIigoGl3EsTBo9gA==","size":{"width":225,"height":40}}

By Property:-

{"font":{"family":"Arial","color":"#000000","size":10},"type":"$","id":"1-1","code":"$\\int_{a}^{b}f\\left(x\\right).dx=\\int_{a}^{b}f\\left(a+b-x\\right).dx$","ts":1602403552089,"cs":"bbUey+1BLemSHcefOSC9vQ==","size":{"width":220,"height":21}}

{"font":{"family":"Arial","size":10,"color":"#000000"},"id":"2-0-0-0-0-0-0-0-0-0-0-0-0-0-1-1-0-1-0","type":"align*","code":"\\begin{align*}\n{I}&={\\int_{0}^{a}\\frac{{\\sqrt[]{a-x}}}{{\\sqrt[]{a-x}}+{\\sqrt[]{a-\\left(a-x\\right)}}}.dx}\\\\\n{I}&={\\int_{0}^{a}\\frac{{\\sqrt[]{a-x}}}{{\\sqrt[]{a-x}}+{\\sqrt[]{x}}}.dx....\\left(2\\right)}\t\n\\end{align*}","ts":1602854044061,"cs":"ufQxpWrxbcReSgVpb7xTHg==","size":{"width":248,"height":89}}

Add eq. 1 and 2:-

{"id":"3-0-0-0-0-0-0-1-0-0-0-0-0-1-0-1-0","font":{"size":10,"color":"#000000","family":"Arial"},"type":"align*","code":"\\begin{align*}\n{2I}&={\\int_{0}^{a}\\frac{{\\sqrt[]{x}}}{{\\sqrt[]{a-x}}+{\\sqrt[]{x}}}.dx+\\int_{0}^{a}\\frac{{\\sqrt[]{a-x}}}{{\\sqrt[]{a-x}}+{\\sqrt[]{x}}}.dx}\\\\\n{2I}&={\\int_{0}^{a}\\frac{{\\sqrt[]{a-x}}+{\\sqrt[]{x}}}{{\\sqrt[]{a-x}}+{\\sqrt[]{x}}}.dx}\\\\\n{2I}&={\\int_{0}^{a}1.dx}\\\\\n{2I}&={\\left[x\\right]_{0}^{a}}\\\\\n{2I}&={a}\\\\\n{I\\,\\,\\,}&={\\frac{a}{2}}\t\n\\end{align*}","ts":1602589109046,"cs":"OASlws6ERShlbEj+sQ27tw==","size":{"width":353,"height":204}}

{"font":{"family":"Arial","color":"#000000","size":10},"type":"$","code":"$18.\\,\\int_{0}^{4}\\left|x-1\\right|.dx$","id":"4-0-0-1-1-1-1-0-1-0","ts":1602589214447,"cs":"jv1jJUyJEAA+/ZmdAsB/Nw==","size":{"width":112,"height":21}}

Solun:- Let f(x) = |x - 1|

Integrate f(x):-

{"font":{"family":"Arial","size":10,"color":"#000000"},"code":"\\begin{align*}\n{I}&={\\int_{0}^{4}\\left|x-1\\right|.dx....\\left(1\\right)}\t\n\\end{align*}","id":"2-0-0-0-0-0-0-0-0-0-0-0-0-0-0-1-1-1-1-0-1","type":"align*","ts":1602589263106,"cs":"tTXsB+JUAaTSh4xkhpDJpg==","size":{"width":168,"height":38}}

By Property:-

{"type":"$","code":"$\\int_{a}^{b}f\\left(x\\right).dx=\\int_{a}^{c}f\\left(x\\right).dx+\\int_{c}^{b}f\\left(x\\right).dx$","font":{"size":10,"family":"Arial","color":"#000000"},"id":"1-2-2","ts":1602493604045,"cs":"i5/s8j6kPci7mHgn0Zayxg==","size":{"width":260,"height":21}}

{"type":"align*","code":"\\begin{align*}\n{I}&={\\int_{0}^{1}-\\left(x-1\\right).dx+\\int_{1}^{4}\\left(x-1\\right).dx}\t\n\\end{align*}","font":{"size":10,"color":"#000000","family":"Arial"},"id":"2-0-0-0-0-0-0-0-0-0-0-0-0-0-1-1-1-1-1-0-1","ts":1602589353176,"cs":"Z6fKBztLd2gRaaPwnq6IXg==","size":{"width":253,"height":38}}

We know that:-

{"type":"$","font":{"color":"#000000","family":"Arial","size":10},"id":"7-0","code":"$\\int_{}^{}x^{n}.dx=\\frac{x^{n+1}}{n+1}+C$","ts":1602493752975,"cs":"ThIogt4GrHQEnxouMS0BCQ==","size":{"width":136,"height":21}}

{"font":{"size":10,"family":"Arial","color":"#000000"},"code":"\\begin{align*}\n{I}&={-\\left[\\frac{x^{2}}{2}-x\\right]_{0}^{1}+\\left[\\frac{x^{2}}{2}-x\\right]_{1}^{4}}\\\\\n{I}&={-\\left[\\left(\\frac{1}{2}-1\\right)-\\left(0\\right)\\right]+\\left[\\left(\\frac{16}{2}-4\\right)-\\left(\\frac{1}{2}-1\\right)\\right]}\\\\\n{I}&={-\\left[-\\frac{1}{2}\\right]+\\left[4+\\frac{1}{2}\\right]}\\\\\n{I}&={5}\t\n\\end{align*}","type":"align*","id":"3-0-0-0-0-0-0-1-0-0-0-0-0-1-2-1-0-1","ts":1602589637417,"cs":"LNGoqWfOpN/CqBm7GmeF6Q==","size":{"width":357,"height":148}}

19. Show that {"code":"$\\int_{0}^{a}f\\left(x\\right).g\\left(x\\right).dx=2\\int_{0}^{a}f\\left(x\\right).dx$","id":"4-0-0-1-1-1-1-0-1-1","type":"$","font":{"size":10,"family":"Arial","color":"#000000"},"ts":1602590181918,"cs":"ypdIIWJoMMHE0TRnT50qFQ==","size":{"width":216,"height":18}}, if f and g are defined as f(x) = f(a - x) and g(x) + g(a - x) = 4.

Solun:- Given {"code":"$\\int_{0}^{a}f\\left(x\\right).g\\left(x\\right).dx=2\\int_{0}^{a}f\\left(x\\right).dx$","id":"17-0","type":"$","font":{"size":10,"family":"Arial","color":"#000000"},"ts":1602590181918,"cs":"817jhlNcuJFlvBx/tVc/dw==","size":{"width":216,"height":18}}

Taking L.H.S.:-

{"id":"18","font":{"color":"#000000","family":"Arial","size":10},"code":"$I=\\int_{0}^{a}f\\left(x\\right).g\\left(x\\right).dx...\\left(1\\right)$","type":"$","ts":1602591110144,"cs":"JgiuPnxjSecvAia1ctWsFQ==","size":{"width":180,"height":18}}

By Property:-

{"font":{"family":"Arial","color":"#000000","size":10},"type":"$","id":"1-1","code":"$\\int_{a}^{b}f\\left(x\\right).dx=\\int_{a}^{b}f\\left(a+b-x\\right).dx$","ts":1602403552089,"cs":"bbUey+1BLemSHcefOSC9vQ==","size":{"width":220,"height":21}}

{"font":{"family":"Arial","color":"#000000","size":10},"type":"align*","code":"\\begin{align*}\n{I}&={\\int_{0}^{a}f\\left(x\\right).g\\left(x\\right).dx}\\\\\n{I}&={\\int_{0}^{a}f\\left(a-x\\right).g\\left(a-x\\right).dx}\\\\\n{I}&={\\int_{0}^{a}f\\left(x\\right).g\\left(a-x\\right).dx....\\left(2\\right)}\t\n\\end{align*}","id":"2-0-0-0-0-0-0-0-0-0-0-0-0-0-1-1-0-1-1","ts":1602591131064,"cs":"mxA/smbYoi9rO/obaAh/WQ==","size":{"width":220,"height":118}}

Add eq. 1 and 2:-

{"code":"\\begin{align*}\n{2I}&={\\int_{0}^{a}f\\left(x\\right).g\\left(x\\right).dx+\\int_{0}^{a}f\\left(x\\right).g\\left(a-x\\right).dx}\\\\\n{2I}&={\\int_{0}^{a}f\\left(x\\right).\\left[g\\left(x\\right)+g\\left(a-x\\right)\\right].dx}\\\\\n{2I}&={\\int_{0}^{a}f\\left(x\\right)\\times4.dx}\\\\\n{2I}&={4\\int_{0}^{a}f\\left(x\\right).dx}\\\\\n{I\\,\\,\\,}&={2\\int_{0}^{a}f\\left(x\\right).dx}\t\n\\end{align*}","type":"align*","id":"3-0-0-0-0-0-0-1-0-0-0-0-0-1-0-1-1","font":{"size":10,"family":"Arial","color":"#000000"},"ts":1602591270096,"cs":"pp5c5wfArMjNzxdhBFMv1A==","size":{"width":316,"height":201}}

{"code":"$\\int_{0}^{a}f\\left(x\\right).g\\left(x\\right).dx=2\\int_{0}^{a}f\\left(x\\right).dx$","id":"17-0","type":"$","font":{"size":10,"family":"Arial","color":"#000000"},"ts":1602590181918,"cs":"817jhlNcuJFlvBx/tVc/dw==","size":{"width":216,"height":18}}

Hence Proved...

Choose the correct answer in Exercises 20 and 21.

20. The value of {"id":"17-1","font":{"family":"Arial","size":10,"color":"#000000"},"code":"$\\int_{\\frac{-\\Pi}{2}}^{\\frac{\\Pi}{2}}\\left(x^{3}+x\\cos x+\\tan^{5}x+1\\right).dx$","type":"$","ts":1602664581477,"cs":"B1yfsLmWeyjmeGHCVmjb1A==","size":{"width":230,"height":29}} is

(A) 0 (B) 2 (C) π (D) 1

Solun:- Let f(x) = x3 + xcos x + tan5x + 1

Integrate f(x):-

{"code":"\\begin{align*}\n{I}&={\\int_{\\frac{-\\Pi}{2}}^{\\frac{\\Pi}{2}}\\left(x^{3}+x\\cos x+\\tan^{5}x+1\\right).dx}\\\\\n{I}&={\\int_{\\frac{-\\Pi}{2}}^{\\frac{\\Pi}{2}}x^{3}.dx+\\int_{\\frac{-\\Pi}{2}}^{\\frac{\\Pi}{2}}x\\cos x.dx+\\int_{\\frac{-\\Pi}{2}}^{\\frac{\\Pi}{2}}\\tan^{5}x.dx+\\int_{\\frac{-\\Pi}{2}}^{\\frac{\\Pi}{2}}1.dx}\t\n\\end{align*}","id":"2-0-0-0-0-0-0-0-0-0-0-0-0-0-0-1-1-1-0-1-0-1-1-1-0-1-0","type":"align*","font":{"family":"Arial","color":"#000000","size":10},"ts":1602665435478,"cs":"EqpMBlXzjn8eBjT1GNYhDQ==","size":{"width":412,"height":94}}

By Property:-

{"font":{"color":"#000000","family":"Arial","size":10},"id":"1-2-1-5-1","code":"\\begin{align*}\n{\\int_{-a}^{a}f\\left(x\\right).dx}&={2\\int_{0}^{a}f\\left(x\\right).dx,if\\,f\\left(-x\\right)=f\\left(x\\right)}\\\\\n{\\int_{-a}^{a}f\\left(x\\right).dx}&={0,if\\,f\\left(-x\\right)=-f\\left(x\\right)}\t\n\\end{align*}","type":"align*","ts":1602503490672,"cs":"ju0dUPVndQfGk7GL29ALLg==","size":{"width":309,"height":78}}

{"font":{"color":"#000000","size":10,"family":"Arial"},"code":"\\begin{align*}\n{I}&={0+0+0+\\int_{\\frac{-\\Pi}{2}}^{\\frac{\\Pi}{2}}1.dx}\\\\\n{I}&={2\\int_{0}^{\\frac{\\Pi}{2}}1.dx}\\\\\n{I}&={2\\left[x\\right]_{0}^{\\frac{\\Pi}{2}}}\\\\\n{I}&={\\frac{2\\Pi}{2}=\\Pi}\t\n\\end{align*}","id":"2-0-0-0-0-0-0-0-0-0-0-0-0-0-0-1-1-1-0-1-0-1-1-1-0-1-1","type":"align*","ts":1602665801042,"cs":"01CVhbubtpNtmcnzNGtwXA==","size":{"width":170,"height":156}}

The correct answer is C.

{"font":{"family":"Arial","color":"#000000","size":10},"id":"4-0-0-1-0-0-1","type":"$","code":"$21.\\,\\int_{0}^{\\frac{\\Pi}{2}}\\log_{}\\left(\\frac{4+3\\sin x}{4+3\\cos x}\\right).dx$","ts":1602666074409,"cs":"X5v0LtzOFlLdzAlKYfujow==","size":{"width":164,"height":25}}

Solun:- Let f(x) = {"code":"$\\log_{}\\frac{4+3\\sin x}{4+3\\cos x}$","font":{"size":10,"color":"#000000","family":"Arial"},"type":"$","id":"15-1","ts":1602666100159,"cs":"agKs/9g4K3igyRkG8veRxw==","size":{"width":76,"height":20}}

Integrate f(x):-

{"font":{"color":"#000000","family":"Arial","size":10},"code":"\\begin{align*}\n{I}&={\\int_{0}^{\\frac{\\Pi}{2}}\\log_{}\\left(\\frac{4+3\\sin x}{4+3\\cos x}\\right).dx....\\left(1\\right)}\t\n\\end{align*}","id":"2-0-0-0-0-0-0-0-0-0-0-0-0-0-0-1-0-0-1","type":"align*","ts":1602666132881,"cs":"KMOmXFRLUCHKzcBvnbfefA==","size":{"width":256,"height":41}}

By Property:-

{"font":{"family":"Arial","color":"#000000","size":10},"type":"$","id":"1-1","code":"$\\int_{a}^{b}f\\left(x\\right).dx=\\int_{a}^{b}f\\left(a+b-x\\right).dx$","ts":1602403552089,"cs":"bbUey+1BLemSHcefOSC9vQ==","size":{"width":220,"height":21}}

{"code":"\\begin{align*}\n{I}&={\\int_{0}^{\\frac{\\Pi}{2}}\\log_{}\\left(\\frac{4+3\\sin \\left(\\frac{\\Pi}{2}-x\\right)}{4+3\\cos \\left(\\frac{\\Pi}{2}-x\\right)}\\right).dx}\\\\\n{I}&={\\int_{0}^{\\frac{\\Pi}{2}}\\log_{}\\left(\\frac{4+3\\cos x}{4+3\\sin x}\\right).dx....\\left(2\\right)}\t\n\\end{align*}","type":"align*","font":{"family":"Arial","color":"#000000","size":10},"id":"2-0-0-0-0-0-0-0-0-0-0-0-0-0-1-1-0-0-1","ts":1602666202234,"cs":"RD9LNhD0lcdsgMj2+Cezmg==","size":{"width":258,"height":93}}

Add eq. 1 and 2:-

{"type":"align*","font":{"size":10,"family":"Arial","color":"#000000"},"code":"\\begin{align*}\n{2I}&={\\int_{0}^{\\frac{\\Pi}{2}}\\log_{}\\left(\\frac{4+3\\sin x}{4+3\\cos x}\\right).dx+\\int_{0}^{\\frac{\\Pi}{2}}\\log_{}\\left(\\frac{4+3\\cos x}{4+3\\sin x}\\right).dx}\\\\\n{2I}&={\\int_{0}^{\\frac{\\Pi}{2}}\\log_{}\\left[\\left(\\frac{4+3\\sin x}{4+3\\cos x}\\right)\\left(\\frac{4+3\\cos x}{4+3\\sin x}\\right)\\right].dx}\\\\\n{2I}&={\\int_{0}^{\\frac{\\Pi}{2}}\\log_{}\\left(\\frac{16+12\\cos x+12\\sin x+9\\sin x\\cos x}{16+12\\sin x+12\\cos x+9\\sin x\\cos x}\\right).dx}\\\\\n{2I}&={\\int_{0}^{\\frac{\\Pi}{2}}\\log_{}1.dx}\\\\\n{2I}&={0}\\\\\n{I\\,\\,}&={0}\t\n\\end{align*}","id":"3-0-0-0-0-0-0-1-0-0-0-0-0-1-0-0-1","ts":1602666439315,"cs":"QVyv3bmuQIR5BEYq3Mn04Q==","size":{"width":414,"height":222}}

The correct answer is C.


Download PDF of Exercise 7.11

See Also:-

Notes of Integrals

Exercise 7.1

Exercise 7.2

Exercise 7.3

Exercise 7.4

Exercise 7.5

Exercise 7.6

Exercise 7.7

Exercise 7.9

Exercise 7.10


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