{"id":28891,"date":"2022-04-15T12:22:02","date_gmt":"2022-04-15T06:52:02","guid":{"rendered":"https:\/\/yanamtakshashila.com\/?p=28891"},"modified":"2024-04-21T20:49:52","modified_gmt":"2024-04-21T15:19:52","slug":"integration-decomposition-method-exercise-problems-with-solutions","status":"publish","type":"post","link":"https:\/\/yanamtakshashila.com\/?p=28891","title":{"rendered":"INTEGRATION &#8211; DECOMPOSITION METHOD EXERCISE 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(5x^2\\ -\\ \\frac{2}{x^3}\\ +\\ \\frac{1}{x}\\ -\\ 3)\\ dx\\ \\hspace{18cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue} {Soln\\ :}\\ \\hspace{18cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\int(5x^2\\ -\\ \\frac{2}{x^3}\\ +\\ \\frac{1}{x}\\ -\\ 3)\\ dx\\ =\\ 5\\int x^2\\ dx\\ -\\ 2\\int x^{-3}\\ dx\\ +\\ \\int \\frac{1}{x}\\ dx\\ -\\  3\\int 1\\ dx\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ 5\\ \\frac{x^3}{3}\\ &#8211; 2 \\frac{x^{-2}}{-2}\\ +\\ log\\ x\\ -\\ 3\\ x\\ +\\ c\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ 5\\ \\frac{x^3}{3}\\ +\\  x^{-2}\\ +\\ log\\ x\\ -\\ 3\\ x\\ +\\ c\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\therefore\\ \\boxed{\\int(5x^2\\ -\\ \\frac{2}{x^3}\\ +\\ \\frac{1}{x}\\ -\\ 3)\\ dx\\ =\\ 5\\ \\frac{x^3}{3}\\ +\\  x^{-2}\\ +\\ log\\ x\\ -\\ 3\\ x\\ +\\ c}\\]<\/div>\n\n\n<p><iframe width=\"790\" height=\"444\" src=\"https:\/\/www.youtube.com\/embed\/R6h6vKNnkog\" title=\"Integration - decomposition - method (excercise) - Part - 1\" 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} {2\\ .}\\ \\color {red} {Evaluate\\ :}\\ \\int(x^2 \\ +\\  cos\\ x)\\ dx\\ \\hspace{18cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue} {Soln\\ :}\\ \\hspace{18cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\int(x^2 \\ +\\  cos\\ x)\\ dx\\ =\\ \\int x^2\\ dx\\ +\\ \\int cos\\ x\\ dx\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \\frac{x^3}{3}\\ +\\ Sin\\ x\\ +\\  c\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\therefore\\ \\boxed{\\int(x^2 \\ +\\  cos\\ x)\\ dx\\ =\\ \\frac{x^3}{3}\\ +\\ Sin\\ x\\ +\\  c}\\]<\/div>\n\n\n<p><iframe width=\"790\" height=\"444\" src=\"https:\/\/www.youtube.com\/embed\/LEpXAJD0D5k\" title=\"Integration - decomposition - method (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(x^2 \\ +\\  Sec^2\\ x)\\ dx\\ \\hspace{18cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue} {Soln\\ :}\\ \\hspace{18cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\int(x^2 \\ +\\  Sec^2\\ x)\\ dx\\ =\\ \\int x^2\\ dx\\ +\\ \\int Sec^2\\ x\\ dx\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \\frac{x^3}{3}\\ +\\ tan\\ x\\ +\\  c\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\therefore\\ \\boxed{\\int(x^2 \\ +\\  Sec^2\\ x)\\ dx\\ =\\ \\frac{x^3}{3}\\ +\\ tan\\ x\\ +\\  c}\\]<\/div>\n\n\n<p><iframe width=\"790\" height=\"444\" src=\"https:\/\/www.youtube.com\/embed\/jscn8jOVSQ8\" title=\"Integration - decomposition - method (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 cot^2 x \\ dx\\ \\hspace{18cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue} {Soln\\ :}\\ \\hspace{18cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\int cot^2 x \\ dx = \\int(cosec^2 x &#8211; 1)\\ dx\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\int cosec^2 x\\ dx &#8211; \\int 1\\ dx\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ -\\ cot\\ x\\ -\\ x\\ +\\ c\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\therefore\\ \\boxed{\\int cot^2 x  \\ dx\\ =\\ -\\ cot\\ x\\ -\\ x\\ +\\ c}\\]<\/div>\n\n\n<p><iframe width=\"790\" height=\"444\" src=\"https:\/\/www.youtube.com\/embed\/mbI7XUEpCfo\" title=\"Integration - decomposition - method (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<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} {5\\ .}\\ \\color {red} {Evaluate\\ :} \\int(x^2 + x + 1) (x\\ +\\ 5)\\ dx\\ \\hspace{18cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue} {Soln\\ :}\\ \\hspace{18cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\int (x^2 + x + 1) (x\\ +\\ 5)\\ dx = \\int(x^3\\ +\\ x^2\\ +\\ x\\ +\\ 5\\ x^2\\ +\\ 5\\ x\\ +\\ 5)\\ dx\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\int (x^3\\ +\\ 6\\ x^2\\ +\\ 6\\ x\\ +\\ 5)\\ dx\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \\frac{x^4}{4}\\ +\\  6 \\frac{x^3}{3}\\ +\\ 6\\ \\frac{x^2}{2}\\ +\\ 5\\ x\\ +\\ c\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \\frac{x^4}{4}\\ +\\  2\\ x^3\\ +\\ 3\\ x^2\\ +\\ 5\\ x\\ +\\ c\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\therefore\\ \\boxed{\\int (x^2 + x + 1) (x\\ +\\ 5)\\ dx\\ =\\ \\frac{x^4}{4}\\ +\\  2\\ x^3\\ +\\ 3\\ x^2\\ +\\ 5\\ x\\ + c}\\]<\/div>\n\n\n<p><iframe width=\"790\" height=\"444\" src=\"https:\/\/www.youtube.com\/embed\/awO2l51pO_o\" title=\"Integration - decomposition - method - Part - 5\" 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} {6\\ .}\\ \\color {red} {Evaluate\\ :} \\int \\frac{sin^2 x}{ 1\\ +\\ cos x} \\ dx\\ \\hspace{18cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue} {Soln\\ :}\\ \\hspace{18cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\int \\frac{sin^2 x}{ 1\\ +\\ cos x} \\ dx = \\int\\frac{1 &#8211; cos^2 x}{ 1\\ +\\ cos x} \\ dx\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= \\int\\frac{(1+ cos x)(1 &#8211; cos x)}{1\\ +\\ cos x}\\ dx\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\int ( 1\\ -\\ cos x)\\ dx\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ x\\ -\\ sin x\\ +\\ c\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\therefore\\ \\boxed{\\int \\frac{sin^2 x}{ 1\\ +\\ cos x} \\ dx\\ =\\ x\\ -\\ sin x\\ +\\ c}\\]<\/div>\n\n\n<p><iframe width=\"790\" height=\"444\" src=\"https:\/\/www.youtube.com\/embed\/ZfHlsknUFv4\" title=\"Integration - decomposition - method - Part - 6\" 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\\ :} \\int \\frac{dx}{1\\ -\\ sin\\ x}\\ \\hspace{18cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue} {Soln\\ :}\\ \\hspace{18cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\int \\frac{dx}{1\\ -\\ sin\\ x} \\ dx = \\int\\frac{dx}{ 1\\ -\\ sin\\ x} \\  \u00d7\\ \\frac{1\\ +\\ sin\\ x}{1\\ -\\ sin\\ x}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= \\int\\frac{1\\ +\\ sin\\ x}{1\\ -\\ sin^2 x}\\ dx\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= \\int\\frac{1\\ +\\ sin\\ x}{cos^2 x}\\ dx\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= \\int\\frac{1}{cos^2 x}\\ dx\\ +\\ \\int\\frac{sin\\ x}{cos^2 x}\\ dx\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= \\int\\ sec^2 x\\ dx\\ +\\ \\int tan\\ x\\ sec\\ x\\ dx\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ tan\\ x\\ +\\ sec\\ x\\ +\\ c\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\therefore\\ \\boxed{\\int \\frac{dx}{ 1\\ -\\ sin\\ x}\\  =\\ tan\\ x\\ +\\ sec\\ x\\ +\\ c}\\]<\/div>\n\n\n<p><iframe width=\"790\" height=\"444\" src=\"https:\/\/www.youtube.com\/embed\/MXcyPSB7eSM\" title=\"Integration - decomposition - method - Part - 7\" 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\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\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- C}}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {8\\ .}\\ \\color {red} {Evaluate\\ :}\\ (i)\\  \\int (\\sqrt{x}\\ +\\ \\frac{1}{\\sqrt{x}}) \\ dx\\ \\hspace{2cm}\\ (ii)\\  \\int cos\\ x\\ cos\\ 12\\ x\\ \\ dx\\ \\hspace{8cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue} {Soln\\ :}\\ \\hspace{18cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[(i)\\ \\int  (\\sqrt{x}\\ +\\ \\frac{1}{\\sqrt{x}}) \\ dx\\ =\\  \\int\\ \\sqrt{x}\\ dx\\ +\\ \\int \\frac{1}{\\sqrt{x}}\\ dx\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= \\int\\ x^{\\frac{1}{2}}\\ dx\\ +\\ \\int x^{\\frac{-1}{2}}\\ dx\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= \\  \\frac{x^{\\frac{3}{2}}}{{\\frac{3}{2}}}\\ +\\ \\frac{x^{\\frac{1}{2}}}{{\\frac{1}{2}}}\\ +\\ c\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= \\  \\frac{2}{3}\\ x^{\\frac{3}{2}}\\ +\\ 2\\ x^{\\frac{1}{2}}\\ +\\ c\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\therefore\\ \\boxed{\\int (\\sqrt{x}\\ +\\ \\frac{1}{\\sqrt{x}})\\ dx\\  =\\ \\frac{2}{3}\\ x^{\\frac{3}{2}}\\ +\\ 2\\ x^{\\frac{1}{2}}\\ +\\ c}\\]<\/div>\n\n\n<p><iframe width=\"790\" height=\"444\" src=\"https:\/\/www.youtube.com\/embed\/DBpYilzmEZM\" title=\"Integration - decomposition - method (Exercise) - Part - 8\" 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   cos\\ x\\ cos\\ 12\\ x\\ dx\\ =\\ \\frac{1}{2}[\\int((cos\\ (x\\ +\\ 12x)\\ +\\ cos(x\\ -\\ 12x))\\ dx]\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= \\frac{1}{2}[\\int(cos\\ 13\\ x\\ +\\ cos\\ (-11x))\\ dx]\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= \\frac{1}{2}[\\int cos\\ 13\\ x\\ dx\\ +\\ \\int cos\\ 11\\ x\\ dx]\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= \\frac{1}{2}[\\frac{sin\\ 13\\ x}{13}\\ +\\ \\frac{sin\\ 11\\ x}{11}]\\ +\\ c\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\therefore\\ \\boxed{\\int cos\\ x\\ cos\\ 12\\ x\\ dx\\  =\\ \\frac{1}{2}[\\frac{sin\\ 13\\ x}{13}\\ +\\ \\frac{sin\\ 11\\ x}{11}]\\ +\\ c}\\]<\/div>\n\n\n<p><iframe width=\"790\" height=\"444\" src=\"https:\/\/www.youtube.com\/embed\/xAnh5di03DY\" title=\"Integration - decomposition - method (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\\ :}\\ (i)\\  \\int (x\\ -\\ 1) (2x\\ +\\ 3)\\ dx\\ \\hspace{2cm}\\ (ii)\\  \\int 2\\ sin\\ 3\\ x\\ cos\\ x\\ \\ dx\\ \\hspace{8cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue} {Soln\\ :}\\ \\hspace{18cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[(i)\\ \\int  (x\\ -\\ 1) (2x\\ +\\ 3) \\ dx\\ =\\ \\int(2\\ x^2\\  +\\ 3\\ x\\  -\\ 2\\ x\\ -\\ 3)\\ dx\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= \\int(2\\ x^2\\ +\\ x\\ -\\ 3)\\ dx\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ =\\ 2 \\frac{x^3}{3}\\ +\\ \\frac{x^2}{2}\\ -\\ 3\\ x + c\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\therefore\\ \\boxed{\\int(x\\ -\\ 1) ( 2x\\ +\\ 3)\\ dx\\ =\\ 2 \\frac{x^3}{3}\\ +\\ \\frac{x^2}{2}\\ -\\ 3\\ x + c}\\]<\/div>\n\n\n<p><iframe width=\"790\" height=\"444\" src=\"https:\/\/www.youtube.com\/embed\/yuiD2wCD9bo\" title=\"Integration - decomposition - method (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 2\\ sin 3x\\  cos x\\ dx\\ =\\ \\int((sin (3x\\ +\\ x)\\ +\\ sin ( 3x\\ -\\ x))\\ dx]\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \\int(sin\\ 4x\\ +\\ sin\\ 2x)\\ dx\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \\int sin 4x\\ dx\\ +\\ \\int sin 2x\\ dx\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ -\\frac{cos\\ 4x}{4}\\ -\\ \\frac{cos\\ 2x}{2}\\ +\\ c\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\therefore\\ \\boxed{\\int 2\\ sin 3x\\  cos x\\ dx\\ =\\ -\\frac{cos\\ 4x}{4}\\ -\\ \\frac{cos\\ 2x}{2}\\ +\\ c}\\]<\/div>\n\n\n<p><iframe width=\"790\" height=\"444\" src=\"https:\/\/www.youtube.com\/embed\/1M2tFiLpcZg\" title=\"Integration - decomposition - method (exercise) - Part - 9 (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} {10.}\\ \\color {red}{Evaluate\\ :} \\int sin 7x\\  sin 3x\\ dx\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {11\\ .}\\ \\color {red} {Evaluate\\ :}\\  \\int sin^3\\ x\\ dx\\ \\hspace{18cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue} {Soln\\ :}\\ \\hspace{18cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\int sin^3\\ x dx\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[W.\\ K.\\ T\\ sin\\ 3\\ x\\ =\\ 3\\ sin\\ x\\ -\\ 4\\ sin^3\\ x\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[sin^3\\ x\\ =\\ \\frac{1}{4}[3\\ sin\\ x\\ -\\ sin\\ 3\\ x]\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\int sin^3\\ x\\ dx\\ =\\ \\int \\frac{1}{4}[3\\ sin\\ x\\ -\\ sin\\ 3\\ x]\\ dx\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\frac{1}{4}[3\\int sin\\ x\\ dx &#8211; \\int sin\\ 3x\\ dx]\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\frac{1}{4}[ -\\ 3\\ cos\\ x\\ +\\  \\frac{cos\\ 3x}{3}]\\ + c\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\therefore\\ \\boxed{\\int sin^3 x \\ dx\\ =\\ \\frac{1}{4}[-\\ 3\\ cos\\ x\\ +\\  \\frac{cos\\ 3x}{3}] + c}\\]<\/div>\n\n\n<p><iframe width=\"790\" height=\"444\" src=\"https:\/\/www.youtube.com\/embed\/gJ0_ijxjHbM\" title=\"Integration - decomposition - method (exercise) - Part - 10\" 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","protected":false},"excerpt":{"rendered":"","protected":false},"author":187055548,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"set","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_memberships_contains_paid_content":false,"footnotes":"","jetpack_publicize_message":"","jetpack_publicize_feature_enabled":true,"jetpack_social_post_already_shared":true,"jetpack_social_options":{"image_generator_settings":{"template":"highway","default_image_id":0,"font":"","enabled":false},"version":2},"_wpas_customize_per_network":false,"jetpack_post_was_ever_published":false},"categories":[711788427],"tags":[],"class_list":["post-28891","post","type-post","status-publish","format-standard","hentry","category-integration-decomposition-method-exercise-problems-with-solutions"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.4 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>INTEGRATION &#8211; 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