{"id":29052,"date":"2022-04-19T16:54:11","date_gmt":"2022-04-19T11:24:11","guid":{"rendered":"https:\/\/yanamtakshashila.com\/?p=29052"},"modified":"2023-03-05T20:59:18","modified_gmt":"2023-03-05T15:29:18","slug":"methods-of-integration-integration-by-substitution-excercise-problems-with-solutions","status":"publish","type":"post","link":"https:\/\/yanamtakshashila.com\/?p=29052","title":{"rendered":"METHODS OF INTEGRATION &#8211; INTEGRATION BY SUBSTITUTION (Excercise Problems with Solutions)"},"content":{"rendered":"\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\LARGE{\\color {purple} {PART- A}}\\]<script src=\"https:\/\/yanamtakshashila.com\/wp-includes\/js\/dist\/hooks.min.js?ver=dd5603f07f9220ed27f1\" id=\"wp-hooks-js\"><\/script>\n<script src=\"https:\/\/yanamtakshashila.com\/wp-includes\/js\/dist\/i18n.min.js?ver=c26c3dc7bed366793375\" id=\"wp-i18n-js\"><\/script>\n<script id=\"wp-i18n-js-after\">\nwp.i18n.setLocaleData( { 'text direction\\u0004ltr': [ 'ltr' ] } );\n\/\/# sourceURL=wp-i18n-js-after\n<\/script>\n<script  async src=\"https:\/\/cdnjs.cloudflare.com\/ajax\/libs\/mathjax\/2.7.7\/MathJax.js?config=TeX-MML-AM_CHTML\" id=\"mathjax-js\"><\/script>\n<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {1\\ .}\\ \\color {red} {Evaluate\\ :}\\ \\int\\ sec^2\\ 5\\ x\\ dx\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Soln:}\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[put\\ u\\ =\\ 5\\ x\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{du}{dx}\\ =\\ 5\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[dx\\ = \\frac{1}{5}\\ du\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\int\\ sec^2\\ 5\\ x\\ dx = \\int sec^2\\ u\\ du\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ tan\\ u\\ +\\ c\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ tan\\ 5\\ x\\ +\\ c\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{\\int\\ sec^2\\ 5\\ x\\ dx\\ =\\ tan\\  5\\ x\\ + c}\\]<\/div>\n\n\n<p><iframe width=\"790\" height=\"444\" src=\"https:\/\/www.youtube.com\/embed\/Vt-JiOCq5J4\" title=\"Integration - Substitution - Exercise - Part - 1\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen=\"\"><\/iframe><\/p>\n\n\n\n<script async=\"\" src=\"https:\/\/pagead2.googlesyndication.com\/pagead\/js\/adsbygoogle.js?client=ca-pub-9453835310745500\" crossorigin=\"anonymous\"><\/script>\n<!-- wide skyscraper -->\n<ins class=\"adsbygoogle\" style=\"display:inline-block;width:300px;height:900px\" data-ad-client=\"ca-pub-9453835310745500\" data-ad-slot=\"9987820756\"><\/ins>\n<script>\n     (adsbygoogle = window.adsbygoogle || []).push({});\n<\/script>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\LARGE{\\color {purple} {PART- B}}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {2\\ .}\\ \\color {red} {Evaluate\\ :}\\ \\int\\ cos^3\\ 7\\ x\\ dx\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Soln:}\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[put\\ u\\ =\\ 7\\ x\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{du}{dx}\\ =\\ 7\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[dx\\ = \\frac{1}{7}\\ du\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\int\\ cos^3\\ 7\\ x\\ dx\\ =\\ \\frac{1}{7}\\int cos^3\\ u\\ du\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cos\\ 3\\ x\\ =\\ 4\\ cos^3\\ x\\ -\\ 3\\ cos\\ x\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cos^3\\ x\\ =\\ \\frac{1}{4}[cos\\ 3\\ x\\ +\\ 3\\ cos\\ x]\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cos^3\\ u\\ =\\ \\frac{1}{4}[cos\\ 3\\ u\\ +\\ 3\\ cos\\ u]\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\int\\ cos^3\\ 7\\ x\\ dx\\ =\\ \\frac{1}{7}\\ \\int\\ \\frac{1}{4}[cos\\ 3\\ u\\ +\\ 3\\ cos\\ u]\\ dx\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \\frac{1}{28}[\\int cos\\ 3\\ u\\ dx\\ +\\ 3\\int\\ cos\\ u\\ dx]\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \\frac{1}{28}[\\frac{sin\\ 3\\ u} {3}\\ +\\ 3\\  sin\\ u]\\ + c\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \\frac{1}{28}[\\frac{sin\\ 3\\ (7\\ x)} {3}\\ +\\ 3\\  sin\\ 7\\ x]\\ + c\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \\frac{1}{28}[\\frac{sin\\ 21\\ x} {3}\\ +\\ 3\\  sin\\ 7\\ x]\\ + c\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{\\int\\ cos^3\\ 7\\ x\\ dx\\ =\\ \\frac{1}{28}[\\frac{sin\\ 21\\ x} {3}\\ +\\ 3\\  sin\\ 7\\ x]\\ + c}\\]<\/div>\n\n\n<p><iframe width=\"790\" height=\"444\" src=\"https:\/\/www.youtube.com\/embed\/lC0inNdWVxA\" title=\"Integration - Substitution - Exercise - Part - 2\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen=\"\"><\/iframe><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {3\\ .}\\ \\color {red} {Evaluate\\ :}\\ \\int\\ \\frac{2x\\ -\\ 1}{{\\sqrt{(x^2\\ -\\ x\\ -\\ 1)}}}\\ dx\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Soln:}\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[put\\   u\\ =\\ x^2\\ -\\ x\\ -\\ 1\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{du}{dx}= \\ 2x\\ -\\ 1\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[du\\ =\\ (2\\ x \\ -\\ 1)\\ dx\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\int\\ \\frac{2x\\ -\\ 1}{{\\sqrt{(x^2\\ -\\ x\\ -\\ 1)}}}\\ dx= \\int\\ \\frac{du}{u}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \\int\\ \\frac{du}{\\sqrt{u}}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\frac{u^{\\frac{-1}{2} + 1}}{\\frac{-1}{2} + 1} + c\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\frac{u^{\\frac{1}{2}}}{\\frac{1}{2}} + c\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=2u^{\\frac{1}{2}} + c\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=2(x^2\\ -\\ x\\ -\\ 1)^{\\frac{1}{2}}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{\\int\\ \\frac{2x\\ -\\ 1}{{\\sqrt{(x^2\\ -\\ x\\ -\\ 1)}}}\\ dx\\ =\\ 2\\ \\sqrt{(x^2\\ -\\ x\\ -\\  1)} + c}\\]<\/div>\n\n\n<p><iframe width=\"790\" height=\"444\" src=\"https:\/\/www.youtube.com\/embed\/WZGrSOTkICQ\" title=\"Integration - Substitution - Exercise - Part - 3\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen=\"\"><\/iframe><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {4\\ .}\\ \\color {red} {Evaluate\\ :}\\ \\int\\frac{sec^2\\ x}{5\\ +\\ 4\\ tan\\ x}\\ dx\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Soln:}\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[put\\   u\\ =\\ 5\\ +\\ 4\\ tan\\ x\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{du}{dx}= \\ 4\\ sec^2\\ x\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{1}{4}\\ du\\ =\\ sec^2\\ x\\ dx\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\int\\frac{sec^2\\ x}{5\\ +\\ 4\\ tan\\ x}\\ dx\\ =\\ \\frac{1}{4} \\int\\ \\frac{du}{u}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \\frac{1}{4}\\ log\\ u\\ + c\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \\frac{1}{4}\\ log(5\\ +\\ 4\\ tan\\ x)\\ + c\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{\\int\\ \\frac{sec^2\\ x}{5\\ +\\ 4\\ tan\\ x}\\ dx\\ =\\ \\frac{1}{4}\\ log(5\\ +\\ 4\\ tan\\ x)\\ + c}\\]<\/div>\n\n\n<p><iframe width=\"790\" height=\"444\" src=\"https:\/\/www.youtube.com\/embed\/hWOBf7vAZ8c\" title=\"Integration - Substitution - Exercise - Part - 4\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen=\"\"><\/iframe><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {5\\ .}\\ \\color {red} {Evaluate\\ :}\\ \\int\\frac{(tan^{-1}\\ x)^3}{1\\ +\\ x^2}\\ dx\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Soln:}\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[put\\   u\\ =\\tan^{-1}\\ x\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{du}{dx}= \\ \\frac{1}{1\\ +\\ x^2}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[du\\ =\\ \\frac{1}{1\\ +\\ x^2}\\ dx\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\int \\frac{(tan^{-1}\\ x)^3}{1\\ +\\ x^2}\\ dx\\ = \\int\\ u^3\\ du\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \\frac{u^4}{4} + c\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \\frac{(tan^{-1}\\ x)^4}{4}\\ +\\ c\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{\\int\\ \\frac{(tan^{-1}\\ x)^3}{1\\ +\\ x^2}\\ dx\\ =\\ \\frac{(tan^{-1}\\ x)^4}{4}\\ + c}\\]<\/div>\n\n\n<p><iframe width=\"790\" height=\"444\" src=\"https:\/\/www.youtube.com\/embed\/lViE-bVtE8A\" title=\"Integration - Substitution - Exercise - Part - 5\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen=\"\"><\/iframe><\/p>\n\n\n\n<script async=\"\" src=\"https:\/\/pagead2.googlesyndication.com\/pagead\/js\/adsbygoogle.js?client=ca-pub-9453835310745500\" crossorigin=\"anonymous\"><\/script>\n<!-- wide skyscraper -->\n<ins class=\"adsbygoogle\" style=\"display:inline-block;width:300px;height:900px\" data-ad-client=\"ca-pub-9453835310745500\" data-ad-slot=\"9987820756\"><\/ins>\n<script>\n     (adsbygoogle = window.adsbygoogle || []).push({});\n<\/script>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\underline{PART\\ -\\ C}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {6\\ .}\\ \\color {red} {Evaluate\\ :}\\ \\hspace{2cm}\\ (i)\\ \\int\\ \\frac{x\\ +\\ 1}{ x^2\\ +\\ 2x\\ -\\ 1}\\ dx\\ \\hspace{2cm}\\ (ii)\\ \\int\\frac{sec^2\\ x}{5\\ +\\  tan\\ x}\\ dx\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Soln:}\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[(i)\\ \\int\\ \\frac{x\\ +\\ 1}{ x^2\\ +\\ 2x\\ -\\ 1}\\ dx\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[put\\   u\\ =\\ x^2\\ +\\ 2x\\ -\\ 1\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{du}{dx}= \\ 2x + 2\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[du\\ =\\ 2(x\\ +\\ 1)\\ dx\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{1}{2}\\ du\\ =\\ (x\\ +\\ 1)\\ dx\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\int\\ \\frac{x+ 1}{x^2\\ +\\ 2x\\ -\\ 1}\\ dx\\ =\\ \\frac{1}{2}\\int\\ \\frac{du}{u}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \\frac{1}{2}\\ log\\ u + c\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \\frac{1}{2}\\ log( x^2\\ +\\ 2x\\ -\\ 1) + c\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{\\int\\ \\frac{x+ 1}{x^2\\ +\\ 2x\\ -\\ 1}\\ dx\\ =\\ \\frac{1}{2}\\ log( x^2\\ +\\ 2x\\ -\\ 1)\\ + c}\\]<\/div>\n\n\n<p><iframe width=\"790\" height=\"444\" src=\"https:\/\/www.youtube.com\/embed\/rV0xyHOEdY8\" title=\"Integration - Substitution - Exercise - Part - 6(i)\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen=\"\"><\/iframe><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[(ii)\\ \\int\\ \\frac{sec^2\\ x}{5\\ +\\ tan\\ x}\\ dx\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[put\\   u\\ =\\ 5\\ +\\  tan\\ x\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{du}{dx}= \\ sec^2\\ x\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[du\\ =\\ sec^2\\ x\\ dx\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\int\\frac{sec^2\\ x}{5\\ +\\  tan\\ x}\\ dx\\ =\\  \\int\\ \\frac{du}{u}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\  log\\ u\\ + c\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\  log(5\\ +\\ tan\\ x)\\ + c\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{\\int\\ \\frac{sec^2\\ x}{5\\ +\\ tan\\ x}\\ dx\\ =\\  log(5\\ +\\ tan\\ x)\\ + c}\\]<\/div>\n\n\n<p><iframe width=\"790\" height=\"444\" src=\"https:\/\/www.youtube.com\/embed\/RXj-RNLOP8o\" title=\"Integration - Substitution - Exercise - Part - 6 (ii)\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen=\"\"><\/iframe><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {7\\ .}\\ \\color {red} {Evaluate\\ :}\\ \\hspace{2cm}\\ (i)\\ \\int\\ \\frac{6x^2\\ -\\ 1}{ 2x^3\\ -\\ x\\ +\\ 5}\\ dx\\ \\hspace{2cm}\\ (ii)\\ \\int\\frac{sin\\ \\sqrt{x}}{\\sqrt{x}}\\ dx\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Soln:}\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[(i)\\ \\int\\ \\frac{6x^2\\ -\\ 1}{ 2x^3\\ -\\ x\\ +\\ 5}\\ dx\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[put\\   u\\ =\\ 2x^3\\ -\\ x\\ +\\ 5\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{du}{dx}= \\ 6x^2\\ -\\ 1\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[du\\ =\\ (6x^2\\ -\\ 1)\\ dx\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\int\\ \\frac{6x^2\\ -\\ 1}{ 2x^3\\ -\\ x\\ +\\ 5}\\ dx\\ =\\ \\int\\ \\frac{du}{u}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\  log\\ u\\ + c\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\  log(2x^3\\ -\\ x\\ +\\ 5)\\ + c\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{\\int\\ \\frac{6x^2\\ -\\ 1}{ 2x^3\\ -\\ x\\ +\\ 5}\\ dx\\ =\\  log(2x^3\\ -\\ x\\ +\\ 5)\\ + c}\\]<\/div>\n\n\n<p><iframe width=\"790\" height=\"444\" src=\"https:\/\/www.youtube.com\/embed\/CLGw8sXpboE\" title=\"Integration - Substitution - Part - 7 (i)\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen=\"\"><\/iframe><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[(ii)\\ \\int\\ \\frac{sin\\ \\sqrt{x}}{\\sqrt{x}}\\ dx\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[put\\   u\\ =\\ \\sqrt{x}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{du}{dx}= \\ \\frac{1}{2\\sqrt{x}}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[2\\ du\\ =\\ \\frac{1}{\\sqrt{x}}\\ dx\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\int\\ \\frac{sin\\ \\sqrt{x}}{\\sqrt{x}}\\ dx\\ =\\ \\int\\ 2\\ sin\\ u\\ du\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\  -\\ 2\\ cos\\ u\\ + c\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\  -\\ 2\\ cos\\ \\sqrt{x}\\ + c\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{\\int\\ \\frac{sin\\ \\sqrt{x}}{\\sqrt{x}}\\ dx\\ =\\   -\\ 2\\ cos\\ \\sqrt{x}\\ + c}\\]<\/div>\n\n\n<p><iframe width=\"790\" height=\"444\" src=\"https:\/\/www.youtube.com\/embed\/Ba7WID-1EGo\" title=\"Integration - Substitution - Exercise - Part - 7 (ii)\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen=\"\"><\/iframe><\/p>\n\n\n\n<script async=\"\" src=\"https:\/\/pagead2.googlesyndication.com\/pagead\/js\/adsbygoogle.js?client=ca-pub-9453835310745500\" crossorigin=\"anonymous\"><\/script>\n<!-- Leader board 1 -->\n<ins class=\"adsbygoogle\" style=\"display:inline-block;width:728px;height:90px\" data-ad-client=\"ca-pub-9453835310745500\" data-ad-slot=\"8769628924\"><\/ins>\n<script>\n     (adsbygoogle = window.adsbygoogle || []).push({});\n<\/script>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {8\\ .}\\ \\color {red} {Evaluate\\ :}\\ \\hspace{2cm}\\ (i)\\ \\int\\ \\frac{cos\\ x}{ (3\\ -\\ 5\\ sin\\ x)^6}\\ dx\\ \\hspace{2cm}\\ (ii)\\ \\int\\frac{e^{tan^{-1}\\ x}}{1\\ +\\ x^2}\\ dx\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Soln:}\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[(i)\\ \\int\\ \\frac{cos\\ x}{ (3\\ -\\ 5\\ sin\\ x)^6}\\ dx\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[put\\   u\\ =\\ 3\\ -\\  5\\ sin\\ x\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{du}{dx}= \\ -\\ 5\\ cos\\ x\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{-1}{5}du\\ =\\ cos\\ x\\ dx\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\int\\ \\frac{cos\\ x}{ (3\\ -\\ 5\\ sin\\ x)^6}\\ dx\\ =\\ \\frac{-1}{5}\\int\\ \\frac{du}{u^6}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \\frac{-1}{5} \\int\\ u^{-6}\\ du\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\frac{-1}{5} (\\frac{u^{-5}}{-5}) + c\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\frac{(3\\ -\\ 5\\ sin\\ x)^{-5}}{25} + c\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{\\int\\ \\frac{cos\\ x}{ (3\\ -\\ 5\\ sin\\ x)^6}\\ dx\\ =\\   \\frac{(3\\ -\\ 5\\ sin\\ x)^{-5}}{25}\\ + c}\\]<\/div>\n\n\n<p><iframe width=\"790\" height=\"444\" src=\"https:\/\/www.youtube.com\/embed\/JJrMm7JmotQ\" title=\"Integration - Substitution - Exercise - Part - 8 (i)\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen=\"\"><\/iframe><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[(ii)\\ \\int\\ \\frac{e^{tan^{-1}\\ x}}{1\\ +\\ x^2}\\ dx\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[put\\   u\\ =\\tan^{-1}\\ x\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{du}{dx}= \\ \\frac{1}{1\\ +\\ x^2}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[du\\ =\\ \\frac{1}{1\\ +\\ x^2}\\ dx\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\int \\frac{e^{tan^{-1}\\ x}}{1\\ +\\ x^2}\\ dx\\ = \\int\\ e^u\\ du\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ e^u + c\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ e^{tan^{-1}\\ x} + c\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{\\int\\ \\frac{e^{tan^{-1}\\ x}}{1\\ +\\ x^2}\\ dx\\ =\\   e^{tan^{-1}\\ x}\\ + c}\\]<\/div>\n\n\n<p><iframe width=\"790\" height=\"444\" src=\"https:\/\/www.youtube.com\/embed\/O--eLcqwOLE\" title=\"Integration - Substitution - Exercise - Part - 8 (ii)\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen=\"\"><\/iframe><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {9\\ .}\\ \\color {red} {Evaluate\\ :}\\ \\hspace{2cm}\\ (i)\\ \\int\\ \\frac{e^x}{1\\ +\\ e^x}\\ dx\\ \\hspace{2cm}\\ (ii)\\ \\int\\ tan^5\\ x\\ sec^2\\ x\\ dx\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Soln:}\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[(i)\\ \\int\\ \\frac{e^x}{1\\ +\\ e^x}\\ dx\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[put\\   u\\ =\\ 1\\ +\\ e^x\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{du}{dx}= \\ e^x\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[du\\ =\\ e^x\\ dx\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\int\\ \\frac{e^x}{1\\ +\\ e^x}\\ dx= \\int\\ \\frac{du}{u}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\  log\\ u\\ + c\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\  log(1\\ +\\ e^x)\\ + c\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{\\int\\ \\frac{e^x}{1\\ +\\ e^x}\\ dx\\ =\\    log(1\\ +\\ e^x)\\ + c}\\]<\/div>\n\n\n<p><iframe width=\"790\" height=\"444\" src=\"https:\/\/www.youtube.com\/embed\/GjLOX2VdA0k\" title=\"Integration - Substitution - Exercise - Part - 9 (i)\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen=\"\"><\/iframe><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[(ii)\\ \\int\\ tan^5\\ x\\ sec^2\\ x\\ dx\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[put\\   u\\ =\\ tan\\ x\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{du}{dx}\\ =\\ sec^2\\ x\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[du\\ =\\ sec^2\\ x\\ dx\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\int\\ tan^5\\ x\\ sec^2\\ x\\ dx = \\int u^5\\ du\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\frac{u^6}{6} + c\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\frac{tan^6 x}{6} + c\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{\\int\\ tan^5\\ x\\ sec^2\\ x\\ dx\\ =\\ \\frac{tan^6 x}{6} + c}\\]<\/div>\n\n\n<p><iframe width=\"1264\" height=\"549\" src=\"https:\/\/www.youtube.com\/embed\/WyX0ArLwZuI\" title=\"Integration - Substitution - Exercise - Part - 9 (ii)\" frameborder=\"0\" allow=\"accelerometer; 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