{"id":13172,"date":"2020-12-10T09:48:38","date_gmt":"2020-12-10T04:18:38","guid":{"rendered":"https:\/\/cbselibrary.com\/?p=13172"},"modified":"2020-12-10T10:39:09","modified_gmt":"2020-12-10T05:09:09","slug":"integration-by-parts","status":"publish","type":"post","link":"https:\/\/cbselibrary.com\/integration-by-parts\/","title":{"rendered":"Integration by Parts"},"content":{"rendered":"<h2 class=\"center\">Integration by Parts<\/h2>\n<p><iframe loading=\"lazy\" title=\"What is Integration by Parts - How to do Integration by Parts\" width=\"1200\" height=\"675\" src=\"https:\/\/www.youtube.com\/embed\/bLhxQIdbWW8?feature=oembed\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen><\/iframe><\/p>\n<p><img loading=\"lazy\" class=\"alignnone size-full wp-image-128720\" src=\"https:\/\/cbselibrary.com\/wp-content\/uploads\/2017\/05\/Integration-by-Parts-1.jpg\" alt=\"Integration by Parts 1\" width=\"701\" height=\"443\" srcset=\"https:\/\/cbselibrary.com\/wp-content\/uploads\/2017\/05\/Integration-by-Parts-1.jpg 701w, https:\/\/cbselibrary.com\/wp-content\/uploads\/2017\/05\/Integration-by-Parts-1-300x190.jpg 300w\" sizes=\"(max-width: 701px) 100vw, 701px\" \/><br \/>\n<strong>(1) When integrand involves more than one type of functions:<\/strong><br \/>\nWe may solve such integrals by a rule which is known as <strong><a href=\"https:\/\/cbselibrary.com\/integration-rules\/\">integration<\/a> by parts<\/strong>. We know that,<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-128721\" src=\"https:\/\/cbselibrary.com\/wp-content\/uploads\/2017\/05\/Integration-by-Parts-2.jpg\" alt=\"Integration by Parts 2\" width=\"426\" height=\"195\" srcset=\"https:\/\/cbselibrary.com\/wp-content\/uploads\/2017\/05\/Integration-by-Parts-2.jpg 426w, https:\/\/cbselibrary.com\/wp-content\/uploads\/2017\/05\/Integration-by-Parts-2-300x137.jpg 300w\" sizes=\"(max-width: 426px) 100vw, 426px\" \/><br \/>\ni.e., the integral of the product of two functions = (First function) \u00d7 (Integral of second function) \u2013 Integral of {(Differentiation of first function) \u00d7 (Integral of second function)}<br \/>\nBefore applying this rule proper choice of first and second function is necessary. Normally we use the following methods :<\/p>\n<ol>\n<li>In the product of two functions, one of the function is not directly integrable (i.e., log|x|, sin<sup>-1<\/sup>x, cos<sup>-1<\/sup>x, tan<sup>-1<\/sup>x&#8230;&#8230;etc), then we take it as the first function and the remaining function is taken as the second function.<\/li>\n<li>If there is no other function, then unity is taken as the second function e.g. in the integration of\u00a0<strong>\u222b<\/strong>sin<sup>-1<\/sup>x dx, <strong>\u222b<\/strong>log<strong>\u00a0<\/strong>x dx, 1 is taken as the second function.<\/li>\n<li>If both of the function are directly integrable then the first function is chosen in such a way that the derivative of the function thus obtained under integral sign is easily integrable. Usually, we use the following preference order for the first function.<br \/>\n(Inverse, Logarithmic, Algebraic, Trigonometric, Exponential). This rule is simply called as \u201c<strong>ILATE<\/strong>\u201d.<\/li>\n<\/ol>\n<p><strong>(2) Integral is of the form \u222be<sup>x<\/sup> {f(x)+f<em>&#8216;<\/em>(x)} dx:<\/strong><br \/>\nIf the integral is of the form <strong>\u222b<\/strong>e<sup>x<\/sup> {f(x)+f<em>&#8216;<\/em>(x)} dx\u00a0then by breaking this integral into two integrals integrate one integral by parts and keeping other integral as it is, by doing so, we get<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-128722\" src=\"https:\/\/cbselibrary.com\/wp-content\/uploads\/2017\/05\/Integration-by-Parts-3.jpg\" alt=\"Integration by Parts 3\" width=\"408\" height=\"162\" srcset=\"https:\/\/cbselibrary.com\/wp-content\/uploads\/2017\/05\/Integration-by-Parts-3.jpg 408w, https:\/\/cbselibrary.com\/wp-content\/uploads\/2017\/05\/Integration-by-Parts-3-300x119.jpg 300w\" sizes=\"(max-width: 408px) 100vw, 408px\" \/><\/p>\n<p><strong>(3) Integral is of the form \u222b[x f<em>&#8216;<\/em>(x)+f(x)]\u00a0dx: <\/strong><br \/>\nIf the integral is of the form <strong>\u222b<\/strong>[x f<em>&#8216;<\/em>(x)+f(x)] dx\u00a0then by breaking this integral into two integrals, integrate one integral by parts and keeping other integral as it is, by doing so, we get,<br \/>\n<strong>\u222b<\/strong>[x f<em>&#8216;<\/em>(x)+f(x)]\u00a0dx = x f(x) + c<\/p>\n<p><strong>(4) Integrals of the form \u222be<sup>ax <\/sup>sin bx dx \u222be<sup>ax <\/sup>cos bx dx:<\/strong><br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-128724\" src=\"https:\/\/cbselibrary.com\/wp-content\/uploads\/2017\/05\/Integration-by-Parts-4.jpg\" alt=\"Integration by Parts 4\" width=\"545\" height=\"524\" srcset=\"https:\/\/cbselibrary.com\/wp-content\/uploads\/2017\/05\/Integration-by-Parts-4.jpg 545w, https:\/\/cbselibrary.com\/wp-content\/uploads\/2017\/05\/Integration-by-Parts-4-300x288.jpg 300w\" sizes=\"(max-width: 545px) 100vw, 545px\" \/><\/p>\n<p><iframe loading=\"lazy\" title=\"Integration by parts - Hard example\" width=\"1200\" height=\"900\" src=\"https:\/\/www.youtube.com\/embed\/QK521aR1X3Y?feature=oembed\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen><\/iframe><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Integration by Parts (1) When integrand involves more than one type of functions: We may solve such integrals by a rule which is known as integration by parts. We know that, i.e., the integral of the product of two functions = (First function) \u00d7 (Integral of second function) \u2013 Integral of {(Differentiation of first function) &#8230; <a title=\"Integration by Parts\" class=\"read-more\" href=\"https:\/\/cbselibrary.com\/integration-by-parts\/\" aria-label=\"More on Integration by Parts\">Read more<\/a><\/p>\n","protected":false},"author":5,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"spay_email":""},"categories":[5],"tags":[4741],"yoast_head":"<!-- This site is optimized with the Yoast SEO Premium plugin v17.8 (Yoast SEO v18.4.1) - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Integration by Parts - CBSE Library<\/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:\/\/cbselibrary.com\/integration-by-parts\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Integration by Parts\" \/>\n<meta property=\"og:description\" content=\"Integration by Parts (1) When integrand involves more than one type of functions: We may solve such integrals by a rule which is known as integration by parts. 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