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Exercise 7.7

Integrate the functions in Exercises 1 to 9.

{"code":"$1.\\,{\\sqrt[]{4-x^{2}}}$","font":{"family":"Arial","size":10,"color":"#000000"},"type":"$","id":"4","ts":1601869038241,"cs":"bH+FNasB10k/MVftvQJIUA==","size":{"width":70,"height":16}}

Solun:- Let f(x) = {"font":{"color":"#000000","size":10,"family":"Arial"},"code":"${\\sqrt[]{4-x^{2}}}$","id":"6","type":"$","ts":1601869181320,"cs":"NndkTlCD+aSroUrExdBJGw==","size":{"width":56,"height":16}}

Integrate f(x):-

{"id":"2-0-0-0-0-0-0-0-0-0-0-0","code":"$\\int_{}^{}f\\left(x\\right)dx=\\int_{}^{}{\\sqrt[]{4-x^{2}}}.dx$","type":"$","font":{"family":"Arial","size":10,"color":"#000000"},"ts":1601869224702,"cs":"GEvFPecUDZ6OdQF2/Stnhg==","size":{"width":176,"height":20}}

We know that:-

{"code":"$\\int_{}^{}{\\sqrt[]{a^{2}-x^{2}}}.dx=\\frac{x}{2}{\\sqrt[]{a^{2}-x^{2}}}+\\frac{a^{2}}{2}\\sin^{-1}\\left(\\frac{x}{a}\\right)+C$","font":{"family":"Arial","size":10,"color":"#000000"},"id":"3-0-0-0-0-0-0-0-0-0","type":"$","ts":1601869356244,"cs":"ETONB1M9abCsdnfSVjiH5g==","size":{"width":332,"height":20}}

{"id":"3-0-0-0-0-0-0-1-0-0-0","code":"\\begin{align*}\n{\\int_{}^{}f\\left(x\\right)dx}&={\\frac{x}{2}{\\sqrt[]{4-x^{2}}}+\\frac{4}{2}\\sin^{-1}\\left(\\frac{x}{2}\\right)+C}\\\\\n{\\int_{}^{}f\\left(x\\right)dx}&={\\frac{x}{2}{\\sqrt[]{4-x^{2}}}+2\\sin^{-1}\\left(\\frac{x}{2}\\right)+C}\t\n\\end{align*}","type":"align*","font":{"size":10,"color":"#000000","family":"Arial"},"ts":1601869472021,"cs":"6wg0iMgcyufF1J9ZROxvNg==","size":{"width":300,"height":76}}

{"font":{"size":10,"color":"#000000","family":"Arial"},"code":"$2.\\,{\\sqrt[]{1-4x^{2}}}$","id":"5-0","type":"$","ts":1601869625351,"cs":"HoKbv+QLzby+ZHP+ACv4bg==","size":{"width":78,"height":16}}

Solun:- Let f(x) = {"code":"${\\sqrt[]{1-4x^{2}}}$","type":"$","font":{"family":"Arial","size":10,"color":"#000000"},"id":"7-0","ts":1601869637405,"cs":"MghFFNac5+QZhEbef/VOtQ==","size":{"width":64,"height":16}}

Integrate f(x):-

{"type":"align*","font":{"color":"#000000","family":"Arial","size":10},"id":"2-0-0-0-0-0-0-0-0-0-0-1-0","code":"\\begin{align*}\n{\\int_{}^{}f\\left(x\\right)dx}&={\\int_{}^{}{\\sqrt[]{1-4x^{2}}}.dx}\\\\\n{\\int_{}^{}f\\left(x\\right)dx}&={\\int_{}^{}{\\sqrt[]{4\\left(\\frac{1}{4}-x^{2}\\right)}}.dx}\\\\\n{\\int_{}^{}f\\left(x\\right)dx}&={2\\int_{}^{}{\\sqrt[]{\\left(\\frac{1}{4}-x^{2}\\right)}}.dx}\t\n\\end{align*}","ts":1601869716253,"cs":"kHutITdPC5s9Oo77a/Uo8w==","size":{"width":232,"height":141}}

We know that:-

{"code":"$\\int_{}^{}{\\sqrt[]{a^{2}-x^{2}}}.dx=\\frac{x}{2}{\\sqrt[]{a^{2}-x^{2}}}+\\frac{a^{2}}{2}\\sin^{-1}\\left(\\frac{x}{a}\\right)+C$","font":{"family":"Arial","size":10,"color":"#000000"},"id":"3-0-0-0-0-0-0-0-0-1","type":"$","ts":1601869356244,"cs":"Ny8N/jz3qMTgqsn0Kq8QkQ==","size":{"width":332,"height":20}}

{"id":"3-0-0-0-0-0-0-1-0-0-1-0","code":"\\begin{align*}\n{\\int_{}^{}f\\left(x\\right)dx}&={2\\left[\\frac{x}{2}{\\sqrt[]{\\frac{1}{4}-x^{2}}}+\\frac{\\frac{1}{4}}{2}\\sin^{-1}\\left(\\frac{x}{\\frac{1}{2}}\\right)\\right]+C}\\\\\n{\\int_{}^{}f\\left(x\\right)dx}&={2\\left[\\frac{x}{2}{\\sqrt[]{\\frac{1-4x^{2}}{4}}}+\\frac{1}{8}\\sin^{-1}\\left(2x\\right)\\right]+C}\\\\\n{\\int_{}^{}f\\left(x\\right)dx}&={\\frac{x}{2}{\\sqrt[]{1-4x^{2}}}+\\frac{1}{4}\\sin^{-1}\\left(2x\\right)+C}\t\n\\end{align*}","font":{"family":"Arial","color":"#000000","size":10},"type":"align*","ts":1601870002096,"cs":"1z9bL3ls/g4dRhsYCSL7qw==","size":{"width":348,"height":140}}

{"code":"$3.\\,{\\sqrt[]{x^{2}+4x+6}}$","type":"$","id":"5-1-0","font":{"color":"#000000","family":"Arial","size":10},"ts":1601870056059,"cs":"mOibqYGDyp4koWiajmoThg==","size":{"width":108,"height":16}}

Solun:- Let f(x) = {"font":{"size":10,"family":"Arial","color":"#000000"},"code":"${\\sqrt[]{x^{2}+4x+6}}$","id":"7-1-0","type":"$","ts":1601870082498,"cs":"PIJTrzgqQQ7Xjd4sipAHdw==","size":{"width":92,"height":16}}

Integrate f(x):-

{"id":"2-0-0-0-0-0-0-0-0-0-0-1-1-0","code":"\\begin{align*}\n{\\int_{}^{}f\\left(x\\right)dx}&={\\int_{}^{}{\\sqrt[]{x^{2}+4x+6}}.dx}\\\\\n{\\int_{}^{}f\\left(x\\right)dx}&={\\int_{}^{}{\\sqrt[]{x^{2}+4x+\\left(2\\right)^{2}-\\left(2\\right)^{2}+6}}.dx}\\\\\n{\\int_{}^{}f\\left(x\\right)dx}&={\\int_{}^{}{\\sqrt[]{\\left(x+2\\right)^{2}+2}}.dx}\\\\\n{\\int_{}^{}f\\left(x\\right)dx}&={\\int_{}^{}{\\sqrt[]{\\left(x+2\\right)^{2}+\\left({\\sqrt[]{2}}\\right)^{2}}}.dx}\t\n\\end{align*}","font":{"family":"Arial","color":"#000000","size":10},"type":"align*","ts":1601870326674,"cs":"h3jjzwH/DEM2pVrntO5+Dg==","size":{"width":318,"height":164}}

We know that:-

{"id":"3-0-0-0-0-0-0-0-0-2-0","code":"$\\int_{}^{}{\\sqrt[]{x^{2}+a^{2}}}.dx=\\frac{x}{2}{\\sqrt[]{x^{2}+a^{2}}}+\\frac{a^{2}}{2}\\log_{}\\left|x+{\\sqrt[]{x^{2}+a^{2}}}\\right|+C$","type":"$","font":{"family":"Arial","color":"#000000","size":10},"ts":1601870421676,"cs":"4VtvZPkZ+USItb3vJU4B3w==","size":{"width":392,"height":22}}

{"font":{"color":"#000000","family":"Arial","size":10},"code":"\\begin{align*}\n{\\int_{}^{}f\\left(x\\right)dx}&={\\frac{x+2}{2}{\\sqrt[]{{\\left(x+2\\right)}^{2}+2}}+\\frac{2}{2}\\log_{}\\left|\\left(x+2\\right)+{\\sqrt[]{{\\left(x+2\\right)}^{2}+2}}\\right|+C}\\\\\n{\\int_{}^{}f\\left(x\\right)dx}&={\\frac{x+2}{2}{\\sqrt[]{x^{2}+4x+6}}+\\log_{}\\left|\\left(x+2\\right)+{\\sqrt[]{x^{2}+4x+6}}\\right|+C}\t\n\\end{align*}","id":"3-0-0-0-0-0-0-1-0-0-1-1-0","type":"align*","ts":1601870796649,"cs":"UFtSS3CU3+MtJ8F68QXsVQ==","size":{"width":496,"height":77}}

{"type":"$","font":{"color":"#000000","size":10,"family":"Arial"},"id":"5-1-1-0","code":"$4.\\,{\\sqrt[]{x^{2}+4x+1}}$","ts":1601870878370,"cs":"jUtUCvkeyF/a8NW/avgWUA==","size":{"width":108,"height":16}}

Solun:- Let f(x) = {"code":"${\\sqrt[]{x^{2}+4x+1}}$","font":{"family":"Arial","size":10,"color":"#000000"},"type":"$","id":"7-1-1-0","ts":1601870988000,"cs":"s6f4HIn1vwFfcLjnn0LkeQ==","size":{"width":92,"height":16}}

Integrate f(x):-

{"id":"2-0-0-0-0-0-0-0-0-0-0-1-1-1-0","code":"\\begin{align*}\n{\\int_{}^{}f\\left(x\\right)dx}&={\\int_{}^{}{\\sqrt[]{x^{2}+4x+1}}.dx}\\\\\n{\\int_{}^{}f\\left(x\\right)dx}&={\\int_{}^{}{\\sqrt[]{x^{2}+4x+\\left(2\\right)^{2}-\\left(2\\right)^{2}+1}}.dx}\\\\\n{\\int_{}^{}f\\left(x\\right)dx}&={\\int_{}^{}{\\sqrt[]{\\left(x+2\\right)^{2}-3}}.dx}\\\\\n{\\int_{}^{}f\\left(x\\right)dx}&={\\int_{}^{}{\\sqrt[]{\\left(x+2\\right)^{2}-\\left({\\sqrt[]{3}}\\right)^{2}}}.dx}\t\n\\end{align*}","font":{"size":10,"family":"Arial","color":"#000000"},"type":"align*","ts":1601871054379,"cs":"O0znDbElYUyJbefsD2+EIQ==","size":{"width":318,"height":164}}

We know that:-

{"font":{"family":"Arial","color":"#000000","size":10},"code":"$\\int_{}^{}{\\sqrt[]{x^{2}-a^{2}}}.dx=\\frac{x}{2}{\\sqrt[]{x^{2}-a^{2}}}-\\frac{a^{2}}{2}\\log_{}\\left|x+{\\sqrt[]{x^{2}-a^{2}}}\\right|+C$","id":"3-0-0-0-0-0-0-0-0-2-1-0","type":"$","ts":1602075623091,"cs":"5wWuRYCP0ywtOO3aapZUUg==","size":{"width":392,"height":22}}

{"id":"3-0-0-0-0-0-0-1-0-0-1-1-1-0","type":"align*","code":"\\begin{align*}\n{\\int_{}^{}f\\left(x\\right)dx}&={\\frac{x+2}{2}{\\sqrt[]{{\\left(x+2\\right)}^{2}-3}}-\\frac{3}{2}\\log_{}\\left|\\left(x+2\\right)+{\\sqrt[]{{\\left(x+2\\right)}^{2}-3}}\\right|+C}\\\\\n{\\int_{}^{}f\\left(x\\right)dx}&={\\frac{x+2}{2}{\\sqrt[]{x^{2}+4x+1}}-\\frac{3}{2}\\log_{}\\left|\\left(x+2\\right)+{\\sqrt[]{x^{2}+4x+1}}\\right|+C}\t\n\\end{align*}","font":{"size":10,"family":"Arial","color":"#000000"},"ts":1602075674878,"cs":"PuwWaWXcZBBsNtHAXH6jhQ==","size":{"width":496,"height":77}}

{"font":{"family":"Arial","color":"#000000","size":10},"type":"$","code":"$5.\\,{\\sqrt[]{1-4x-x^{2}}}$","id":"5-1-1-1-0","ts":1601885359113,"cs":"ZS7D+nGVrVnoZLA4tnUqNQ==","size":{"width":108,"height":16}}

Solun:- Let f(x) = {"font":{"color":"#000000","family":"Arial","size":10},"id":"7-1-1-1-0","type":"$","code":"${\\sqrt[]{1-4x-x^{2}}}$","ts":1601885389898,"cs":"mgetZdY0szV370/AwuX5yQ==","size":{"width":92,"height":16}}

Integrate f(x):-

{"code":"\\begin{align*}\n{\\int_{}^{}f\\left(x\\right)dx}&={\\int_{}^{}{\\sqrt[]{1-4x-x^{2}}}.dx}\\\\\n{\\int_{}^{}f\\left(x\\right)dx}&={\\int_{}^{}{\\sqrt[]{-\\left(x^{2}+4x-1\\right)}}.dx}\\\\\n{\\int_{}^{}f\\left(x\\right)dx}&={\\int_{}^{}{\\sqrt[]{-\\left(x^{2}+4x+\\left(2\\right)^{2}-\\left(2\\right)^{2}-1\\right)}}.dx}\\\\\n{\\int_{}^{}f\\left(x\\right)dx}&={\\int_{}^{}{\\sqrt[]{-\\left(\\left(x+2\\right)^{2}-5\\right)}}.dx}\\\\\n{\\int_{}^{}f\\left(x\\right)dx}&={\\int_{}^{}{\\sqrt[]{\\left({\\sqrt[]{5}}\\right)^{2}-\\left(x+2\\right)^{2}}}.dx}\t\n\\end{align*}","font":{"size":10,"color":"#000000","family":"Arial"},"type":"align*","id":"2-0-0-0-0-0-0-0-0-0-0-1-1-1-1-0","ts":1601885575978,"cs":"EeVAqH6UQVzkvBy64nIJzw==","size":{"width":349,"height":212}}

We know that:-

{"code":"$\\int_{}^{}{\\sqrt[]{a^{2}-x^{2}}}.dx=\\frac{x}{2}{\\sqrt[]{a^{2}-x^{2}}}+\\frac{a^{2}}{2}\\sin^{-1}\\left(\\frac{x}{a}\\right)+C$","id":"3-0-0-0-0-0-0-0-0-2-1-1-0","font":{"family":"Arial","color":"#000000","size":10},"type":"$","ts":1601885787319,"cs":"KUY8aPYzvwAn6UDH1xOAng==","size":{"width":332,"height":20}}

{"font":{"family":"Arial","color":"#000000","size":10},"code":"\\begin{align*}\n{\\int_{}^{}f\\left(x\\right)dx}&={\\frac{x+2}{2}{\\sqrt[]{5-{\\left(x+2\\right)}^{2}}}+\\frac{5}{2}\\sin^{-1}\\left(\\frac{x+2}{{\\sqrt[]{5}}}\\right)+C}\\\\\n{\\int_{}^{}f\\left(x\\right)dx}&={\\frac{x+2}{2}{\\sqrt[]{1-4x-x^{2}}}+\\frac{5}{2}\\sin^{-1}\\left(\\frac{x+2}{{\\sqrt[]{5}}}\\right)+C}\t\n\\end{align*}","type":"align*","id":"3-0-0-0-0-0-0-1-0-0-1-1-1-1-0","ts":1601886049258,"cs":"4f1xWbytJJj/0rsNdPH70Q==","size":{"width":398,"height":82}}

{"code":"$6.\\,{\\sqrt[]{x^{2}+4x-5}}$","id":"5-1-1-1-1-0","type":"$","font":{"family":"Arial","color":"#000000","size":10},"ts":1601886140189,"cs":"Ho1NewB+f9TEv/ysn20UxQ==","size":{"width":108,"height":16}}

Solun:- Let f(x) = {"code":"${\\sqrt[]{x^{2}+4x-5}}$","type":"$","id":"7-1-1-1-1-0","font":{"size":10,"color":"#000000","family":"Arial"},"ts":1601886155697,"cs":"x4ToIoFjTNk4W53nB+43eQ==","size":{"width":92,"height":16}}

Integrate f(x):-

{"type":"align*","code":"\\begin{align*}\n{\\int_{}^{}f\\left(x\\right)dx}&={\\int_{}^{}{\\sqrt[]{x^{2}+4x-5}}.dx}\\\\\n{\\int_{}^{}f\\left(x\\right)dx}&={\\int_{}^{}{\\sqrt[]{x^{2}+4x+\\left(2\\right)^{2}-\\left(2\\right)^{2}-5}}.dx}\\\\\n{\\int_{}^{}f\\left(x\\right)dx}&={\\int_{}^{}{\\sqrt[]{\\left(x+2\\right)^{2}-9}}.dx}\\\\\n{\\int_{}^{}f\\left(x\\right)dx}&={\\int_{}^{}{\\sqrt[]{\\left(x+2\\right)^{2}-\\left(3\\right)^{2}}}.dx}\t\n\\end{align*}","id":"2-0-0-0-0-0-0-0-0-0-0-1-1-1-1-1-0","font":{"color":"#000000","family":"Arial","size":10},"ts":1601895655302,"cs":"xHnnWhm1v90TjbYkxC0Urw==","size":{"width":318,"height":160}}

We know that:-

{"font":{"color":"#000000","family":"Arial","size":10},"code":"$\\int_{}^{}{\\sqrt[]{x^{2}-a^{2}}}.dx=\\frac{x}{2}{\\sqrt[]{x^{2}-a^{2}}}-\\frac{a^{2}}{2}\\log_{}\\left|x+{\\sqrt[]{x^{2}-a^{2}}}\\right|+C$","id":"3-0-0-0-0-0-0-0-0-2-1-1-1-0","type":"$","ts":1601887028377,"cs":"wOurwYbNOlJg45Tiz9EdHA==","size":{"width":392,"height":22}}

{"id":"3-0-0-0-0-0-0-1-0-0-1-1-1-1-1-0","font":{"size":10,"color":"#000000","family":"Arial"},"code":"\\begin{align*}\n{\\int_{}^{}f\\left(x\\right)dx}&={\\frac{x+2}{2}{\\sqrt[]{{\\left(x+2\\right)}^{2}-{\\left(3\\right)}^{2}}}-\\frac{9}{2}\\log_{}\\left|\\left(x+2\\right)+{\\sqrt[]{{\\left(x+2\\right)}^{2}-{\\left(3\\right)}^{2}}}\\right|+C}\\\\\n{\\int_{}^{}f\\left(x\\right)dx}&={\\frac{x+2}{2}{\\sqrt[]{x^{2}+4x-5}}-\\frac{9}{2}\\log_{}\\left|\\left(x+2\\right)+{\\sqrt[]{x^{2}+4x-5}}\\right|+C}\t\n\\end{align*}","type":"align*","ts":1601887373219,"cs":"q8MP/rBzDuFH/UhooiYYcg==","size":{"width":533,"height":77}}

{"code":"$7.\\,{\\sqrt[]{1+3x-x^{2}}}$","type":"$","id":"5-1-1-1-1-1-0","font":{"size":10,"family":"Arial","color":"#000000"},"ts":1601887480266,"cs":"QGAxB1IaSJGddoorynZzcg==","size":{"width":108,"height":16}}

Solun:- Let f(x) = {"id":"7-1-1-1-1-1-0","code":"${\\sqrt[]{1+3x-x^{2}}}$","type":"$","font":{"family":"Arial","size":10,"color":"#000000"},"ts":1601887523875,"cs":"UAwq2DRB/+6yxaFXHrWaJA==","size":{"width":92,"height":16}}

Integrate f(x):-

{"font":{"size":10,"color":"#000000","family":"Arial"},"code":"\\begin{align*}\n{\\int_{}^{}f\\left(x\\right)dx}&={\\int_{}^{}{\\sqrt[]{1+3x-x^{2}}}.dx}\\\\\n{\\int_{}^{}f\\left(x\\right)dx}&={\\int_{}^{}{\\sqrt[]{-\\left(x^{2}-3x-1\\right)}}.dx}\\\\\n{\\int_{}^{}f\\left(x\\right)dx}&={\\int_{}^{}{\\sqrt[]{-\\left(x^{2}-3x+\\left(\\frac{3}{2}\\right)^{2}-\\left(\\frac{3}{2}\\right)^{2}-1\\right)}}.dx}\\\\\n{\\int_{}^{}f\\left(x\\right)dx}&={\\int_{}^{}{\\sqrt[]{-\\left(\\left(x-\\frac{3}{2}\\right)^{2}-\\frac{13}{4}\\right)}}.dx}\\\\\n{\\int_{}^{}f\\left(x\\right)dx}&={\\int_{}^{}{\\sqrt[]{\\left(\\frac{{\\sqrt[]{13}}}{2}\\right)^{2}-\\left(x-\\frac{3}{2}\\right)^{2}}}.dx}\t\n\\end{align*}","id":"2-0-0-0-0-0-0-0-0-0-0-1-1-1-1-1-1-0","type":"align*","ts":1601888892366,"cs":"79kiZh2Itb4Z6rUUP5gA/Q==","size":{"width":392,"height":249}}

We know that:-

{"id":"3-0-0-0-0-0-0-0-0-2-1-1-1-1-0","font":{"size":10,"color":"#000000","family":"Arial"},"type":"$","code":"$\\int_{}^{}{\\sqrt[]{a^{2}-x^{2}}}.dx=\\frac{x}{2}{\\sqrt[]{a^{2}-x^{2}}}+\\frac{a^{2}}{2}\\sin^{-1}\\left(\\frac{x}{a}\\right)+C$","ts":1601888966452,"cs":"U3dUorGx+XAj2f/yZrjcMg==","size":{"width":332,"height":20}}

{"code":"\\begin{align*}\n{\\int_{}^{}f\\left(x\\right)dx}&={\\frac{x-\\frac{3}{2}}{2}{\\sqrt[]{{\\left(\\frac{{\\sqrt[]{13}}}{2}\\right)}^{2}-{\\left(x-\\frac{3}{2}\\right)}^{2}}}+\\frac{\\frac{13}{4}}{2}\\sin^{-1}\\left(\\frac{x-\\frac{3}{2}}{\\frac{{\\sqrt[]{13}}}{2}}\\right)+C}\\\\\n{\\int_{}^{}f\\left(x\\right)dx}&={\\frac{2x-3}{4}{\\sqrt[]{1+3x-x^{2}}}+\\frac{13}{8}\\sin^{-1}\\left(\\frac{2x-3}{{\\sqrt[]{13}}}\\right)+C}\t\n\\end{align*}","font":{"color":"#000000","family":"Arial","size":10},"type":"align*","id":"3-0-0-0-0-0-0-1-0-0-1-1-1-1-1-1-0","ts":1601890100852,"cs":"TOrm7lKvveRhq86FuAaNPg==","size":{"width":498,"height":100}}

{"type":"$","font":{"color":"#000000","size":10,"family":"Arial"},"code":"$8.\\,{\\sqrt[]{x^{2}+3x}}$","id":"5-1-1-1-1-1-1-0","ts":1601890452065,"cs":"oeAeJjfyPGWqR3SjfmzizA==","size":{"width":80,"height":16}}

Solun:- Let f(x) = {"type":"$","font":{"color":"#000000","size":10,"family":"Arial"},"code":"${\\sqrt[]{x^{2}+3x}}$","id":"7-1-1-1-1-1-1-0","ts":1601890784161,"cs":"2wgtOQ0gemmpM3OexfDSfg==","size":{"width":64,"height":16}}

Integrate f(x):-

{"id":"2-0-0-0-0-0-0-0-0-0-0-1-1-1-1-1-1-1-0-0","type":"align*","font":{"color":"#000000","family":"Arial","size":10},"code":"\\begin{align*}\n{\\int_{}^{}f\\left(x\\right)dx}&={\\int_{}^{}{\\sqrt[]{x^{2}+3x}}.dx}\\\\\n{\\int_{}^{}f\\left(x\\right)dx}&={\\int_{}^{}{\\sqrt[]{x^{2}+3x+\\left(\\frac{3}{2}\\right)^{2}-\\left(\\frac{3}{2}\\right)^{2}}}.dx}\\\\\n{\\int_{}^{}f\\left(x\\right)dx}&={\\int_{}^{}{\\sqrt[]{\\left(x+\\frac{3}{2}\\right)^{2}-\\left(\\frac{3}{2}\\right)^{2}}}.dx}\t\n\\end{align*}","ts":1601890870257,"cs":"YmUBtpAsGMiffMJqU+4UHA==","size":{"width":328,"height":141}}

We know that:-

{"id":"3-0-0-0-0-0-0-0-0-2-1-1-1-1-1-0","type":"$","code":"$\\int_{}^{}{\\sqrt[]{x^{2}-a^{2}}}.dx=\\frac{x}{2}{\\sqrt[]{x^{2}-a^{2}}}-\\frac{a^{2}}{2}\\log_{}\\left|x+{\\sqrt[]{x^{2}-a^{2}}}\\right|+C$","font":{"size":10,"color":"#000000","family":"Arial"},"ts":1601891166024,"cs":"4O4dUX4DleOKh+CEnodWXg==","size":{"width":392,"height":22}}

{"code":"\\begin{align*}\n{\\int_{}^{}f\\left(x\\right)dx}&={\\frac{x+\\frac{3}{2}}{2}{\\sqrt[]{{\\left(x+\\frac{3}{2}\\right)}^{2}-{\\left(\\frac{3}{2}\\right)}^{2}}}-\\frac{\\frac{9}{4}}{2}\\log_{}\\left|\\left(x+\\frac{3}{2}\\right)+{\\sqrt[]{{\\left(x+\\frac{3}{2}\\right)}^{2}-{\\left(\\frac{3}{2}\\right)}^{2}}}\\right|+C}\\\\\n{\\int_{}^{}f\\left(x\\right)dx}&={\\frac{2x+3}{4}{\\sqrt[]{x^{2}+3x}}-\\frac{9}{8}\\log_{}\\left|\\left(x+\\frac{3}{2}\\right)+{\\sqrt[]{x^{2}+3x}}\\right|+C}\t\n\\end{align*}","id":"3-0-0-0-0-0-0-1-0-0-1-1-1-1-1-1-1-0","type":"align*","font":{"color":"#000000","family":"Arial","size":8},"ts":1601891656834,"cs":"yACw0CEVdBHpbgiMUjYH6g==","size":{"width":541,"height":84}}

{"font":{"family":"Arial","size":10,"color":"#000000"},"id":"5-1-1-1-1-1-1-1-0","code":"$9.\\,{\\sqrt[]{1+\\frac{x^{2}}{9}}}$","type":"$","ts":1601891771012,"cs":"ISW+Jl8U7yXDtkO450p4Ag==","size":{"width":76,"height":28}}

Solun:- Let f(x) = {"font":{"family":"Arial","color":"#000000","size":10},"code":"${\\sqrt[]{1+\\frac{x^{2}}{9}}}$","id":"7-1-1-1-1-1-1-1-0","type":"$","ts":1601891800851,"cs":"rF5e4GUMNMYED45inGgaUA==","size":{"width":60,"height":28}}

Integrate f(x):-

{"type":"align*","id":"2-0-0-0-0-0-0-0-0-0-0-1-1-1-1-1-1-1-1-0","font":{"family":"Arial","size":10,"color":"#000000"},"code":"\\begin{align*}\n{\\int_{}^{}f\\left(x\\right)dx}&={\\int_{}^{}{\\sqrt[]{1+\\frac{x^{2}}{9}}}.dx}\\\\\n{\\int_{}^{}f\\left(x\\right)dx}&={\\int_{}^{}{\\sqrt[]{\\frac{1}{9}\\left(9+x^{2}\\right)}}.dx}\\\\\n{\\int_{}^{}f\\left(x\\right)dx}&={\\frac{1}{3}\\int_{}^{}{\\sqrt[]{\\left(9+x^{2}\\right)}}.dx}\\\\\n{\\int_{}^{}f\\left(x\\right)dx}&={\\frac{1}{3}\\int_{}^{}{\\sqrt[]{\\left(x\\right)^{2}+\\left(3\\right)^{2}}}.dx}\t\n\\end{align*}","ts":1601891992509,"cs":"NSmTuGYnP31oOxAzrDBZKA==","size":{"width":238,"height":168}}

We know that:-

{"code":"$\\int_{}^{}{\\sqrt[]{x^{2}+a^{2}}}.dx=\\frac{x}{2}{\\sqrt[]{x^{2}+a^{2}}}+\\frac{a^{2}}{2}\\log_{}\\left|x+{\\sqrt[]{x^{2}+a^{2}}}\\right|+C$","type":"$","font":{"color":"#000000","size":10,"family":"Arial"},"id":"3-0-0-0-0-0-0-0-0-2-1-1-1-1-1-1-0-0","ts":1601892028613,"cs":"uL2/ex7OgEqy0Nz5yrzl0g==","size":{"width":392,"height":22}}

{"type":"align*","id":"3-0-0-0-0-0-0-1-0-0-1-1-1-1-1-1-1-1-0","code":"\\begin{align*}\n{\\int_{}^{}f\\left(x\\right)dx}&={\\frac{1}{3}\\left[\\frac{x}{2}{\\sqrt[]{{\\left(x\\right)}^{2}+{\\left(3\\right)}^{2}}}+\\frac{9}{2}\\log_{}\\left|x+{\\sqrt[]{x^{2}+9}}\\right|\\right]+C}\\\\\n{\\int_{}^{}f\\left(x\\right)dx}&={\\frac{1}{3}\\left[\\frac{x}{2}{\\sqrt[]{{x}^{2}+9}}+\\frac{9}{2}\\log_{}\\left|x+{\\sqrt[]{x^{2}+9}}\\right|\\right]+C}\\\\\n{\\int_{}^{}f\\left(x\\right)dx}&={\\frac{x}{6}{\\sqrt[]{{x}^{2}+9}}+\\frac{3}{2}\\log_{}\\left|x+{\\sqrt[]{x^{2}+9}}\\right|+C}\t\n\\end{align*}","font":{"family":"Arial","size":10,"color":"#000000"},"ts":1601892336616,"cs":"8e/1CidKVGQVO7kDP3oxWQ==","size":{"width":412,"height":120}}

Choose the correct answer in Exercises 10 to 11.

{"id":"5-1-1-1-1-1-1-1-1-0","code":"$10.\\,\\int_{}^{}{\\sqrt[]{1+x^{2}}}.dx$","font":{"color":"#000000","family":"Arial","size":10},"type":"$","ts":1601892752514,"cs":"vD+iD9GAduOsFCz7IfK+Nw==","size":{"width":120,"height":20}}

Solun:- Let f(x) = {"font":{"family":"Arial","size":10,"color":"#000000"},"id":"7-1-1-1-1-1-1-1-1-0","type":"$","code":"${\\sqrt[]{1+x^{2}}}$","ts":1601892772560,"cs":"L9ce5AeJYNhVe19m0RNHmA==","size":{"width":56,"height":16}}

Integrate f(x):-

{"type":"align*","code":"\\begin{align*}\n{\\int_{}^{}f\\left(x\\right)dx}&={\\int_{}^{}{\\sqrt[]{1+x^{2}}}.dx}\\\\\n{\\int_{}^{}f\\left(x\\right)dx}&={\\int_{}^{}{\\sqrt[]{\\left(x^{2}+\\left(1\\right)^{2}\\right)}}.dx}\t\n\\end{align*}","font":{"size":10,"family":"Arial","color":"#000000"},"id":"2-0-0-0-0-0-0-0-0-0-0-1-1-1-1-1-1-1-1-1-0","ts":1601892881326,"cs":"PpTFULxNHdw79PPDUx9vRg==","size":{"width":228,"height":80}}

We know that:-

{"code":"$\\int_{}^{}{\\sqrt[]{x^{2}+a^{2}}}.dx=\\frac{x}{2}{\\sqrt[]{x^{2}+a^{2}}}+\\frac{a^{2}}{2}\\log_{}\\left|x+{\\sqrt[]{x^{2}+a^{2}}}\\right|+C$","type":"$","font":{"color":"#000000","size":10,"family":"Arial"},"id":"3-0-0-0-0-0-0-0-0-2-1-1-1-1-1-1-0-1","ts":1601892028613,"cs":"6/No+mPe3A0qjLahn00wHQ==","size":{"width":392,"height":22}}

{"code":"\\begin{align*}\n{\\int_{}^{}f\\left(x\\right)dx}&={\\frac{x}{2}{\\sqrt[]{{x}^{2}+{\\left(1\\right)}^{2}}}+\\frac{1}{2}\\log_{}\\left|x+{\\sqrt[]{{x}^{2}+{\\left(1\\right)}^{2}}}\\right|+C}\\\\\n{\\int_{}^{}f\\left(x\\right)dx}&={\\frac{x}{2}{\\sqrt[]{{x}^{2}+1}}+\\frac{1}{2}\\log_{}\\left|x+{\\sqrt[]{x^{2}+1}}\\right|+C}\t\n\\end{align*}","type":"align*","font":{"color":"#000000","size":10,"family":"Arial"},"id":"3-0-0-0-0-0-0-1-0-0-1-1-1-1-1-1-1-1-1-0","ts":1601893245943,"cs":"HxoFudetNmV5xqVmrg1Ltw==","size":{"width":386,"height":77}}

The correct answer is A.

{"id":"5-1-1-1-1-1-1-1-1-1","type":"$","code":"$11.\\,\\int_{}^{}{\\sqrt[]{x^{2}-8x+7}}.dx$","font":{"family":"Arial","size":10,"color":"#000000"},"ts":1601893353307,"cs":"R22Zqzxf79jHvJZYGdgLCg==","size":{"width":156,"height":20}}

Solun:- Let f(x) = {"font":{"color":"#000000","size":10,"family":"Arial"},"type":"$","code":"${\\sqrt[]{x^{2}-8x+7}}$","id":"7-1-1-1-1-1-1-1-1-1","ts":1601893380944,"cs":"xX3QYAqZml5Y/dSKhdae3Q==","size":{"width":92,"height":16}}

Integrate f(x):-

{"code":"\\begin{align*}\n{\\int_{}^{}f\\left(x\\right)dx}&={\\int_{}^{}{\\sqrt[]{x^{2}-8x+7}}.dx}\\\\\n{\\int_{}^{}f\\left(x\\right)dx}&={\\int_{}^{}{\\sqrt[]{x^{2}-8x+\\left(4\\right)^{2}-\\left(4\\right)^{2}+7}}.dx}\\\\\n{\\int_{}^{}f\\left(x\\right)dx}&={\\int_{}^{}{\\sqrt[]{\\left(x-4\\right)^{2}-\\left(3\\right)^{2}}}.dx}\t\n\\end{align*}","id":"2-0-0-0-0-0-0-0-0-0-0-1-1-1-1-1-1-1-0-1","type":"align*","font":{"size":10,"color":"#000000","family":"Arial"},"ts":1601893486367,"cs":"xZoldjgEIV5K+DwKGrBRBA==","size":{"width":318,"height":118}}

We know that:-

{"id":"3-0-0-0-0-0-0-0-0-2-1-1-1-1-1-0","type":"$","code":"$\\int_{}^{}{\\sqrt[]{x^{2}-a^{2}}}.dx=\\frac{x}{2}{\\sqrt[]{x^{2}-a^{2}}}-\\frac{a^{2}}{2}\\log_{}\\left|x+{\\sqrt[]{x^{2}-a^{2}}}\\right|+C$","font":{"size":10,"color":"#000000","family":"Arial"},"ts":1601891166024,"cs":"4O4dUX4DleOKh+CEnodWXg==","size":{"width":392,"height":22}}

{"type":"align*","code":"\\begin{align*}\n{\\int_{}^{}f\\left(x\\right)dx}&={\\frac{x-4}{2}{\\sqrt[]{{\\left(x-4\\right)}^{2}-9}}-\\frac{9}{2}\\log_{}\\left|\\left(x-4\\right)+{\\sqrt[]{{\\left(x-4\\right)}^{2}-9}}\\right|+C}\\\\\n{\\int_{}^{}f\\left(x\\right)dx}&={\\frac{x-4}{2}{\\sqrt[]{x^{2}-8x+7}}-\\frac{9}{2}\\log_{}\\left|\\left(x-4\\right)+{\\sqrt[]{x^{2}-8x+7}}\\right|+C}\t\n\\end{align*}","font":{"size":10,"color":"#000000","family":"Arial"},"id":"3-0-0-0-0-0-0-1-0-0-1-1-1-1-1-1-1-1-1-1","ts":1601893731739,"cs":"cfO7w8ldFjHQltVzGHrhYg==","size":{"width":496,"height":77}}

The correct answer is D.


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