{"id":14166,"date":"2021-04-18T07:42:02","date_gmt":"2021-04-18T02:12:02","guid":{"rendered":"https:\/\/yanamtakshashila.com\/?p=14166"},"modified":"2024-05-15T17:15:53","modified_gmt":"2024-05-15T11:45:53","slug":"5-1-integration-by-parts","status":"publish","type":"post","link":"https:\/\/yanamtakshashila.com\/?p=14166","title":{"rendered":"4.1 INTEGRATION BY PARTS"},"content":{"rendered":"\n<div class=\"wp-block-mathml-mathmlblock\">\\[Integrals\\ of\\ the\\ form\\ \\int x\\ sin\\ nx\\ dx,\\  \\int x\\ cos\\ nx\\ dx,\\ \\int x\\ e^{nx}\\ dx,\\]<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\">\\[\\int x^n\\ log\\ x\\ dx\\ and\\ \\int log\\ x\\ dx\\ Simple\\ Problems\\]<\/div>\n\n\n\n<script async=\"\" src=\"https:\/\/pagead2.googlesyndication.com\/pagead\/js\/adsbygoogle.js\"><\/script>\n<!-- sidebar ad 1 -->\n<ins class=\"adsbygoogle\" style=\"display:inline-block;width:326px;height:280px\" data-ad-client=\"ca-pub-9453835310745500\" data-ad-slot=\"6703350399\"><\/ins>\n<script>\n     (adsbygoogle = window.adsbygoogle || []).push({});\n<\/script>\n\n\n\n<p><strong>Introduction:<\/strong><\/p>\n\n\n\n<p class=\"has-text-align-justify\">When the integrand is a product of two functions and the method of decomposition or substitution can not be applied, then the <strong>method of by parts<\/strong> is used.<\/p>\n\n\n\n<script async=\"\" src=\"https:\/\/pagead2.googlesyndication.com\/pagead\/js\/adsbygoogle.js\"><\/script>\n<ins class=\"adsbygoogle\" style=\"display:block; text-align:center;\" data-ad-layout=\"in-article\" data-ad-format=\"fluid\" data-ad-client=\"ca-pub-9453835310745500\" data-ad-slot=\"2812384453\"><\/ins>\n<script>\n     (adsbygoogle = window.adsbygoogle || []).push({});\n<\/script>\n\n\n\n<p><strong>Integraiton by parts formula:<\/strong><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\int u\\ dv = uv &#8211; \\int v\\ du\\]<\/div>\n\n\n\n<p class=\"has-text-align-justify\">The above formula is used by taking proper choice of &#8216;u&#8217; and &#8216;dv&#8217;. &#8216;u&#8217; should be chosen based on thefollowing order of Preference. Simply remember <strong>ILATE<\/strong><\/p>\n\n\n\n<p>1. Inverse trigonometric functions: <\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[{sin}^{-1}x,\\  {cos}^{-1}x,\\ {tan}^{-1}x,\\ etc\\]<\/div>\n\n\n\n<p>2. Logarithmic functions:  log x<\/p>\n\n\n\n<p>3. Algebraic functions:  <\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[1,\\ x\\ x^2,\\ x^3\\ etc\\]<\/div>\n\n\n\n<p>4. Trigonometric functions:  sin x, cos x, tan x, etc.<\/p>\n\n\n\n<p>5. Exponential functions:   <\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[e^x,\\ e^{2x},\\ e^{3x}\\ etc\\] <\/div>\n\n\n\n<script async=\"\" src=\"https:\/\/pagead2.googlesyndication.com\/pagead\/js\/adsbygoogle.js\"><\/script>\n<!-- Ad1 -->\n<ins class=\"adsbygoogle\" style=\"display:block\" data-ad-client=\"ca-pub-9453835310745500\" data-ad-slot=\"8240817448\" data-ad-format=\"auto\" data-full-width-responsive=\"true\"><\/ins>\n<script>\n     (adsbygoogle = window.adsbygoogle || []).push({});\n<\/script>\n\n\n\n<!-- admitad.banner: 9a1p2gsrb0109003a68856637026d8 Redmagic WW -->\n<a target=\"_blank\" rel=\"nofollow noopener\" href=\"https:\/\/ad.admitad.com\/g\/9a1p2gsrb0109003a68856637026d8\/?i=4\"><img fetchpriority=\"high\" decoding=\"async\" width=\"300\" height=\"250\" border=\"0\" src=\"https:\/\/ad.admitad.com\/b\/9a1p2gsrb0109003a68856637026d8\/\" alt=\"Redmagic WW\"><\/a>\n<!-- \/admitad.banner -->\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 1\\ .}\\ \\color {red}{Evaluate\\ :}\\ \\int x\\ sin\\ 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<p class=\"has-text-align-center\"><strong>ILATE<\/strong><\/p>\n\n\n\n<p class=\"has-text-align-center\">u = x&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; dv&nbsp; =&nbsp;sin x dx<\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{du}{dx} = \\frac{d}{dx} (x)\\ \\hspace{2cm}\\ \\int dv = \\int sinx\\ dx\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{du}{dx} = 1\\ \\hspace{2cm}\\  v =- cos x\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\ du = dx\\ \\hspace{5cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\int u\\ dv = uv &#8211; \\int v\\ du\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\int x\\ sin\\ x\\ dx = x ( &#8211; cos x) &#8211; \\int &#8211; cos x\\ dx\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= -\\ x cos x\\ +\\ sin x\\ + c\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{\\int \\ sin\\ x\\ dx = -\\ x cos x\\ +\\ sin x\\ +c}\\]<\/div>\n\n\n<p><iframe width=\"790\" height=\"444\" src=\"https:\/\/www.youtube.com\/embed\/7nDDaX9ytT4\" title=\"Integration By Parts - 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<ins class=\"adsbygoogle\" style=\"display:block\" data-ad-format=\"autorelaxed\" data-ad-client=\"ca-pub-9453835310745500\" data-ad-slot=\"6771622407\"><\/ins>\n<script>\n     (adsbygoogle = window.adsbygoogle || []).push({});\n<\/script>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 2\\ .}\\ \\color {red}{Evaluate\\ :}\\ \\int x\\ e^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<p class=\"has-text-align-center\"> <strong>ILATE<\/strong> <\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[u= x\\ \\hspace{2cm}\\  dv = e^x\\ dx\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{du}{dx} = \\frac{d}{dx} (x)\\ \\hspace{2cm}\\ \\int dv = \\int e^x\\ dx\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{du}{dx} = 1\\ \\hspace{2cm}\\  v = e^x\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\ du = dx\\ \\hspace{5cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\int u\\ dv = uv &#8211; \\int v\\ du\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\int x\\ e^x dx = x\\ e^x &#8211; \\int e^x\\ dx\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = x\\ e^x &#8211; e^x +c\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{\\int x\\ e^x dx = x\\ e^x &#8211; e^x +c}\\]<\/div>\n\n\n<p><iframe width=\"790\" height=\"444\" src=\"https:\/\/www.youtube.com\/embed\/Mxxz0CBoBF4\" title=\"Integration By Parts - 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} {Example\\ 3\\ .}\\ \\color {red}{Evaluate\\ :}\\ \\int x\\ e^{3x}\\ dx\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{18cm}\\ April\\ 2024\\]<\/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\">\\[ ILATE \\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[u= x\\ \\hspace{2cm}\\  dv = e^{3x}\\ dx\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{du}{dx} = \\frac{d}{dx} (x)\\ \\hspace{2cm}\\ \\int dv = \\int e^{3x}\\ dx\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{du}{dx} = 1\\ \\hspace{2cm}\\  v = \\frac{e^{3x}}{3}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\ du = dx\\ \\hspace{5cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\int u\\ dv = uv &#8211; \\int v\\ du\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\int x\\ e^{3x} dx = x\\ \\frac{e^{3x}}{3} &#8211; \\int \\frac{e^{3x}}{3}\\ dx\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= x\\ \\frac{e^{3x}}{3} &#8211;  \\frac{e^{3x}}{9}\\ + c\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{\\int x\\ e^{3x}\\ dx = x\\ \\frac{e^{3x}}{3} &#8211;  \\frac{e^{3x}}{9} +c}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 4\\ .}\\ \\color {red}{Evaluate\\ :}\\ \\int x\\ e^{-\\ 7x}\\ dx\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{18cm}\\ April\\ 2024\\]<\/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\">\\[ ILATE \\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[u= x\\ \\hspace{2cm}\\  dv = e^{-7x}\\ dx\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{du}{dx} = \\frac{d}{dx} (x)\\ \\hspace{2cm}\\ \\int dv = \\int e^{-7x}\\ dx\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{du}{dx} = 1\\ \\hspace{2cm}\\  v = \\frac{e^{-7x}}{-7}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\ du = dx\\ \\hspace{5cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\int u\\ dv = uv &#8211; \\int v\\ du\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\int x\\ e^{-7x} dx = x\\ \\frac{e^{-7x}}{-7} &#8211; \\int \\frac{e^{-7x}}{-7}\\ dx\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=-\\ x\\ \\frac{e^{-7x}}{7} +\\ \\frac{1}{7}  \\frac{e^{-7x}}{-7}\\  + c\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{\\int x\\ e^{-7x} dx = -\\ x\\ \\frac{e^{-7x}}{7} &#8211;  \\frac{e^{-7x}}{49} +c}\\]<\/div>\n\n\n<p><iframe width=\"790\" height=\"444\" src=\"https:\/\/www.youtube.com\/embed\/VFrNtF1HZt8\" title=\"Integration By Parts - 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} {Example\\ 5\\ .}\\ \\color {red}{Evaluate\\ :}\\ \\int x\\ e^{nx}\\ dx\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/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\">\\[ ILATE \\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[u= x\\ \\hspace{2cm}\\  dv = e^{nx}\\ dx\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{du}{dx} = \\frac{d}{dx} (x)\\ \\hspace{2cm}\\ \\int dv = \\int e^{nx}\\ dx\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{du}{dx} = 1\\ \\hspace{2cm}\\  v = \\frac{e^{nx}}{n}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\ du = dx\\ \\hspace{5cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\int u\\ dv = uv &#8211; \\int v\\ du\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\int x\\ e^{nx} dx = x\\ \\frac{e^{nx}}{n} &#8211; \\int \\frac{e^{nx}}{n}\\ dx\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= x\\ \\frac{e^{nx}}{n} &#8211;  \\frac{e^{nx}}{n^2}\\  + c\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{\\int x\\ e^{nx}\\ dx = x\\ \\frac{e^{nx}}{n} &#8211;  \\frac{e^{nx}}{n^2} +c}\\]<\/div>\n\n\n<p><iframe width=\"790\" height=\"444\" src=\"https:\/\/www.youtube.com\/embed\/fTzA_dxeltM\" title=\"Integration By Parts - 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\"><\/div>\n\n\n\n<script async=\"\" src=\"https:\/\/pagead2.googlesyndication.com\/pagead\/js\/adsbygoogle.js\"><\/script>\n<ins class=\"adsbygoogle\" style=\"display:block\" data-ad-format=\"fluid\" data-ad-layout-key=\"-6t+ed+2i-1n-4w\" data-ad-client=\"ca-pub-9453835310745500\" data-ad-slot=\"9770958327\"><\/ins>\n<script>\n     (adsbygoogle = window.adsbygoogle || []).push({});\n<\/script>\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<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<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<ins class=\"adsbygoogle\" style=\"display:block\" data-ad-format=\"autorelaxed\" data-ad-client=\"ca-pub-9453835310745500\" data-ad-slot=\"6771622407\"><\/ins>\n<script>\n     (adsbygoogle = window.adsbygoogle || []).push({});\n<\/script>\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<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<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 6\\ .}\\ \\color {red}{Evaluate\\ :}\\ \\int x\\ sin\\ 2x\\ dx\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Soln:}\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<p class=\"has-text-align-center\"> <strong>ILATE<\/strong>  <\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[u= x\\ \\hspace{2cm}\\  dv =sin\\ 2x\\ dx\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{du}{dx} = \\frac{d}{dx} (x)\\ \\hspace{2cm}\\ \\int dv = \\int sin\\ 2x\\ dx\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{du}{dx} = 1\\ \\hspace{2cm}\\  v = -\\frac{cos\\ 2x}{2}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\ du = dx\\ \\hspace{5cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\int u\\ dv = uv &#8211; \\int v\\ du\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\int x\\ sin\\ 2x\\ dx = &#8211; x\\ \\frac{cos\\ 2x}{2}  + \\int \\frac{cos\\ 2x}{2}\\ dx\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = &#8211; x\\ \\frac{cos\\ 2x}{2}  + \\frac{1}{2}(\\frac{sin\\ 2x}{2})\\ + c\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = &#8211; x\\ \\frac{cos\\ 2x}{2} +\\frac{sin\\ 2x}{4}\\ + c\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{\\int x\\ sin\\ 2x\\ dx = &#8211; x\\ \\frac{cos\\ 2x}{2} +\\frac{sin\\ 2x}{4}\\ + c}\\]<\/div>\n\n\n<p><iframe width=\"790\" height=\"444\" src=\"https:\/\/www.youtube.com\/embed\/at-eNj9Iksw\" title=\"Integration By Parts - 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} {Example\\ 7\\ .}\\ \\color {red}{Evaluate\\ :}\\ \\int x\\ cos\\ 5x\\ dx\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Soln:}\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<p class=\"has-text-align-center\"> <strong>ILATE<\/strong> <\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[u= x\\ \\hspace{2cm}\\  dv =cos\\ 5x\\ dx\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{du}{dx} = \\frac{d}{dx} (x)\\ \\hspace{2cm}\\ \\int dv = \\int cos\\ 5x\\ dx\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{du}{dx} = 1\\ \\hspace{2cm}\\  v = \\frac{sin\\ 5x}{5}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\ du = dx\\ \\hspace{5cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\int u\\ dv = uv &#8211; \\int v\\ du\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\int x\\ cos\\ 5x\\ dx = x\\ \\frac{sin\\ 5x}{5} &#8211; \\int \\frac{sin\\ 5x}{5}\\ dx\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = x\\ \\frac{sin\\ 5x}{5} &#8211; \\frac{1}{5}(\\frac{cos\\ 5x}{5})\\ + c\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = x\\ \\frac{sin\\ 5x}{5} +\\frac{cos\\ 5x}{25}\\ + c\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{\\int x\\ cos\\ 5x\\ dx = x\\ \\frac{sin\\ 5x}{5} +\\frac{cos\\ 5x}{25}\\ + c}\\]<\/div>\n\n\n<p><iframe width=\"790\" height=\"444\" src=\"https:\/\/www.youtube.com\/embed\/Z5m7ByB-URM\" title=\"Integration By Parts - Part - 6\" 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<ins class=\"adsbygoogle\" style=\"display:block\" data-ad-format=\"autorelaxed\" data-ad-client=\"ca-pub-9453835310745500\" data-ad-slot=\"6771622407\"><\/ins>\n<script>\n     (adsbygoogle = window.adsbygoogle || []).push({});\n<\/script>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 8\\ .}\\ \\color {red}{Evaluate\\ :}\\ \\int log\\ x\\ dx\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{18cm}\\ April\\ 2024\\]<\/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\">\\[\\int log\\ x\\ dx = \\int 1.\\ log\\ x\\ dx\\]<\/div>\n\n\n\n<p class=\"has-text-align-center\"> <strong>ILATE<\/strong>  <\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[u= log\\ x\\ \\hspace{2cm}\\  dv = 1\\ dx\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{du}{dx} = \\frac{d}{dx} ( log\\ x)\\ \\hspace{2cm}\\ \\int dv = \\int 1\\ dx\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{du}{dx} = \\frac{1}{x}\\ \\hspace{2cm}\\  v = x\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\ du = \\frac{1}{x}\\ dx\\ \\hspace{5cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\int u\\ dv = uv &#8211; \\int v\\ du\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\int log\\ x\\ dx = x\\ log\\ x- \\int  x\\ (\\frac{1}{x})\\ dx\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\  x\\ log\\ x- \\int  1\\ dx\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\  x\\ log\\ x- x + c\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{\\int log\\ x\\ dx = x\\ log\\ x- x +c}\\]<\/div>\n\n\n<p><iframe width=\"790\" height=\"444\" src=\"https:\/\/www.youtube.com\/embed\/sLgzncTxqx0\" title=\"Integration By Parts - 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\">\\[\\color {purple} {Example\\ 9\\ .}\\ \\color {red}{Evaluate\\ :}\\ \\int\\ x\\ log\\ 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<p class=\"has-text-align-center\"> <strong>ILATE<\/strong>   <\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[u= log\\ x\\ \\hspace{2cm}\\  dv = x\\ dx\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{du}{dx} = \\frac{d}{dx} ( log\\ x)\\ \\hspace{2cm}\\ \\int dv = \\int x\\ dx\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{du}{dx} = \\frac{1}{x}\\ \\hspace{2cm}\\  v = \\frac{x^2}{2}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\ du = \\frac{1}{x}\\ dx\\ \\hspace{5cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\int u\\ dv = uv &#8211; \\int v\\ du\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\int x\\ log\\ x\\ dx =  log\\ x\\ \\frac{x^2}{2} &#8211; \\int  \\frac{x^2}{2}\\ (\\frac{1}{x})\\ dx\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ =  log\\ x\\ \\frac{x^2}{2} &#8211; \\frac{1}{2}\\ \\int  x\\  dx\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ =  log\\ x\\ \\frac{x^2}{2} &#8211; \\frac{1}{2}\\ \\frac{x^2}{2}\\  + c\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ =  log\\ x\\ \\frac{x^2}{2} &#8211; \\frac{x^2}{4}\\  + c\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{\\int x\\ log\\ x\\ dx = log\\ x\\ \\frac{x^2}{2} &#8211; \\frac{x^2}{4}\\ + c}\\]<\/div>\n\n\n<p><iframe width=\"790\" height=\"444\" src=\"https:\/\/www.youtube.com\/embed\/AlQOEMfkMqk\" title=\"Integration By Parts - 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\">\\[\\color {purple} {Example\\ 10\\ .}\\ \\color {red}{Evaluate\\ :}\\ \\int x^2\\ log\\ 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<p class=\"has-text-align-center\"> <strong>ILATE<\/strong>   <\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[u= log\\ x\\ \\hspace{2cm}\\  dv = x^2\\ dx\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{du}{dx} = \\frac{d}{dx} ( log\\ x)\\ \\hspace{2cm}\\ \\int dv = \\int x^2\\ dx\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{du}{dx} = \\frac{1}{x}\\ \\hspace{2cm}\\  v = \\frac{x^3}{3}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\ du = \\frac{1}{x}\\ dx\\ \\hspace{5cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\int u\\ dv = uv &#8211; \\int v\\ du\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\int x^2\\ log\\ x\\ dx =  log\\ x\\ \\frac{x^3}{3} &#8211; \\int  \\frac{x^3}{3}\\ (\\frac{1}{x})\\ dx\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ =  log\\ x\\ \\frac{x^3}{3} &#8211; \\frac{1}{3}\\ \\int  x^2\\  dx\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ =  log\\ x\\ \\frac{x^3}{3} &#8211; \\frac{1}{3}\\ \\frac{x^3}{3}\\  + c\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ =  log\\ x\\ \\frac{x^3}{3} &#8211; \\frac{x^3}{9}\\  + c\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{\\int x^2\\ log\\ x\\ dx = log\\ x\\ \\frac{x^3}{3} &#8211; \\frac{x^3}{9}\\ + c}\\]<\/div>\n\n\n<p><iframe width=\"790\" height=\"444\" src=\"https:\/\/www.youtube.com\/embed\/DKHHFKQTD1U\" title=\"Integration By Parts - Part - 9\" 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<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} {Example\\ 11\\ .}\\ \\color {red}{Evaluate\\ :}\\ \\int x^n\\ log\\ x \\ dx\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace {18cm}\\ April\\ 2024\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Soln:}\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<p class=\"has-text-align-center\"> <strong>ILATE<\/strong>    <\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[u= log\\ x\\ \\hspace{2cm}\\  dv = x^n\\ dx\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{du}{dx} = \\frac{d}{dx} ( log\\ x)\\ \\hspace{2cm}\\ \\int dv = \\int x^n\\ dx\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{du}{dx} = \\frac{1}{x}\\ \\hspace{2cm}\\  v = \\frac{x^{n+1}}{n+1}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\ du = \\frac{1}{x}\\ dx\\ \\hspace{5cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\int u\\ dv = uv &#8211; \\int v\\ du\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\int x^n\\ log\\ x\\ dx =  log\\ x\\ \\frac{x^{n+1}}{n+1} &#8211; \\int  \\frac{x^{n+1}}{n+1}\\ (\\frac{1}{x})\\ dx\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ =  log\\ x\\ \\frac{x^{n+1}}{n+1} &#8211; \\frac{1}{n+1}\\ \\int  x^{n}\\  dx\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ =  log\\ x\\ \\frac{x^{n+1}}{n+1} &#8211; \\frac{1}{(n+1)}\\ \\frac{x^{n+1}}{(n+1)}\\  + c\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ =  log\\ x\\ \\frac{x^{n+1}}{n+1} &#8211; \\frac{1}{(n+1)^2}\\ x^{n+1}\\  + c\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{\\int\\ x^n\\ log\\ x\\ dx = log\\ x\\ \\frac{x^{n+1}}{n+1} &#8211; \\frac{1}{(n+1)^2}\\ x^{n+1}\\ + c}\\]<\/div>\n\n\n<p><iframe width=\"790\" height=\"444\" src=\"https:\/\/www.youtube.com\/embed\/uziinmdHiWs\" title=\"Integration By Parts - Part - 10\" 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} {Example\\ 12\\ .}\\ \\color {red}{Evaluate\\ :}\\ \\int (x\\ +\\ 3)\\ sin\\ 7x\\ dx\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace {18cm}\\ April\\ 2024\\]<\/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\">\\[\\int (x\\ +\\ 3)\\ sin\\ 7x\\ dx\\ =\\ \\int x\\ sin7x\\ dx\\ +\\ 3\\ \\int sin\\ 7x\\ dx\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ =\\ I_1\\ +\\ I_2\\ \\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[I_1\\ =\\ \\ \\int\\ x\\ sin\\ 7x\\ dx\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ ILATE \\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[u= x\\ \\hspace{2cm}\\  dv =sin\\ 7x\\ dx\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{du}{dx} = \\frac{d}{dx} (x)\\ \\hspace{2cm}\\ \\int dv = \\int sin\\ 7x\\ dx\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{du}{dx} = 1\\ \\hspace{2cm}\\  v = -\\frac{cos\\ 7x}{7}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\ du = dx\\ \\hspace{5cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\int u\\ dv = uv &#8211; \\int v\\ du\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\int x\\ sin\\ 7x\\ dx = &#8211; x\\ \\frac{cos\\ 7x}{7}  + \\int \\frac{cos\\ 7x}{7}\\ dx\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = &#8211; x\\ \\frac{cos\\ 7x}{7}  + \\frac{1}{7}(\\frac{sin\\ 7x}{7})\\ + c\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = &#8211; x\\ \\frac{cos\\ 7x}{7} +\\frac{sin\\ 7x}{49}\\ + c\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{\\int x\\ sin\\ 7x\\ dx = &#8211; x\\ \\frac{cos\\ 7x}{7} +\\frac{sin\\ 7x}{49}\\ + c}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[I_2\\ =\\ 3\\ \\int\\  sin\\ 7x\\ dx\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = &#8211; 3\\ \\frac{cos\\ 7x}{7}\\ + c\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{3\\int\\ sin\\ 7x\\ dx\\ =\\ &#8211; 3\\ \\frac{cos\\ 7x}{7}\\ + c}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\int (x\\ +\\ 3)\\ sin\\ 7x\\ dx\\ =\\ &#8211; x\\ \\frac{cos\\ 7x}{7}\\ +\\ \\frac{sin\\ 7x}{49}\\ &#8211; 3\\ \\frac{cos\\ 7x}{7}\\ + c\\]<\/div>\n\n\n<p><iframe width=\"790\" height=\"444\" src=\"https:\/\/www.youtube.com\/embed\/vmqu5pGPlnU\" title=\"Integration By Parts - Part - 11\" 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\"><\/script>\n<ins class=\"adsbygoogle\" style=\"display:block\" data-ad-format=\"autorelaxed\" data-ad-client=\"ca-pub-9453835310745500\" data-ad-slot=\"3040040190\"><\/ins>\n<script>\n     (adsbygoogle = window.adsbygoogle || []).push({});\n<\/script>\n\n\n\n<script async=\"\" src=\"https:\/\/pagead2.googlesyndication.com\/pagead\/js\/adsbygoogle.js?client=ca-pub-9453835310745500\" crossorigin=\"anonymous\"><\/script>\n<!-- Ad1 -->\n<ins class=\"adsbygoogle\" style=\"display:block\" data-ad-client=\"ca-pub-9453835310745500\" data-ad-slot=\"8240817448\" data-ad-format=\"auto\" data-full-width-responsive=\"true\"><\/ins>\n<script>\n     (adsbygoogle = window.adsbygoogle || []).push({});\n<\/script>\n","protected":false},"excerpt":{"rendered":"<p>Introduction: When the integrand is a product of two functions and the method of decomposition or substitution can not be applied, then the method of by parts is used. Integraiton by parts formula: The above formula is used by taking proper choice of &#8216;u&#8217; and &#8216;dv&#8217;. &#8216;u&#8217; should be chosen based on thefollowing order of [&hellip;]<\/p>\n","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":[711787904,711788436],"tags":[],"class_list":["post-14166","post","type-post","status-publish","format-standard","hentry","category-n-integration-by-parts","category-n-unit-iv-integral-calculus-ii"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.4 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>4.1 INTEGRATION BY PARTS - YANAMTAKSHASHILA<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/yanamtakshashila.com\/?p=14166\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"4.1 INTEGRATION BY PARTS - YANAMTAKSHASHILA\" \/>\n<meta property=\"og:description\" content=\"Introduction: When the integrand is a product of two functions and the method of decomposition or substitution can not be applied, then the method of by parts is used. 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