{"id":13157,"date":"2020-12-10T07:55:00","date_gmt":"2020-12-10T02:25:00","guid":{"rendered":"https:\/\/www.aplustopper.com\/?p=13157"},"modified":"2020-12-10T10:36:40","modified_gmt":"2020-12-10T05:06:40","slug":"integration-rules","status":"publish","type":"post","link":"https:\/\/www.aplustopper.com\/integration-rules\/","title":{"rendered":"Integration Rules and Formulas"},"content":{"rendered":"<h2>Integration Rules and Formulas<\/h2>\n<p><iframe loading=\"lazy\" width=\"500\" height=\"375\" src=\"https:\/\/www.youtube.com\/embed\/lxx_i0zCxrU?feature=oembed\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen><\/iframe><\/p>\n<h3>Integral of\u00a0a Function<\/h3>\n<p>A function \u03d5(x) is called a primitive or an antiderivative of a function f(x), if <strong>?'(x)\u00a0= f(x)<\/strong>.<br \/>\nLet f(x) be a function. Then the collection of all its primitives is called the indefinite integral of f(x) and is denoted by <strong>\u222b<\/strong>f(x) dx.<br \/>\nThus,<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-128711\" src=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2020\/01\/Integration-Rules-and-Formulas-1.jpg\" alt=\"Integration Rules and Formulas 1\" width=\"364\" height=\"61\" srcset=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2020\/01\/Integration-Rules-and-Formulas-1.jpg 364w, https:\/\/www.aplustopper.com\/wp-content\/uploads\/2020\/01\/Integration-Rules-and-Formulas-1-300x50.jpg 300w\" sizes=\"(max-width: 364px) 100vw, 364px\" \/><br \/>\nwhere \u03d5(x) is primitive of f(x) and c is an arbitrary constant known as the constant of integration.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-128712\" src=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2020\/01\/Integration-Rules-and-Formulas-2.jpg\" alt=\"Integration Rules and Formulas 2\" width=\"616\" height=\"463\" srcset=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2020\/01\/Integration-Rules-and-Formulas-2.jpg 616w, https:\/\/www.aplustopper.com\/wp-content\/uploads\/2020\/01\/Integration-Rules-and-Formulas-2-300x225.jpg 300w\" sizes=\"(max-width: 616px) 100vw, 616px\" \/><\/p>\n<h3>Integration Rules<\/h3>\n<ol>\n<li><strong>Chain rule :<\/strong><br \/>\n<strong>\u222b<\/strong><em>u.v dx<\/em> = <em>uv<sub>1<\/sub> \u2013 u&#8217;v<sub>2<\/sub> + u&#8221;v<sub>3<\/sub> \u2013 u&#8221;&#8217;v<sub>4<\/sub><\/em> + \u2026\u2026\u2026 + (\u20131)<em><sup>n\u00ad\u20131<\/sup> u<sup>n\u20131<\/sup>v<sub>n<\/sub><\/em> + (\u20131)<em><sup>n<\/sup> <strong>\u222b<\/strong>u<sup>n<\/sup>.v<sub>n<\/sub> dx<\/em><br \/>\nWhere \u00a0stands for <em>n<\/em><sup>th<\/sup> differential coefficient of <em>u<\/em> and stands for <em>n<\/em><sup>th<\/sup> integral of <em>v<\/em>.<\/li>\n<li><strong>Sum Rule<\/strong><br \/>\n<strong>\u222b<\/strong><em>(f + g) dx =\u00a0<strong>\u222b<\/strong>f dx + <strong>\u222b<\/strong>g dx<\/em><\/li>\n<li><strong>Difference Rule<\/strong><br \/>\n<em><strong>\u222b<\/strong>(f &#8211; g) dx =\u00a0<strong>\u222b<\/strong>f dx &#8211; <strong>\u222b<\/strong>g dx<\/em><\/li>\n<li><strong>Multiplication by constant<\/strong><br \/>\n<em><strong>\u222b<\/strong>cf(x) dx = c<strong>\u222b<\/strong>f(x) dx<\/em><\/li>\n<li><strong>Power Rule (n\u2260-1)<\/strong><br \/>\n<em><strong>\u222b<\/strong>x<sup>n<\/sup>\u00a0dx = x<sup>n+1<\/sup>\/(n+1) + C<\/em><\/li>\n<\/ol>\n<h3>Fundamental Integration Formulae<\/h3>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-128713\" src=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2020\/01\/Integration-Rules-and-Formulas-3.jpg\" alt=\"Integration Rules and Formulas 3\" width=\"500\" height=\"586\" srcset=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2020\/01\/Integration-Rules-and-Formulas-3.jpg 500w, https:\/\/www.aplustopper.com\/wp-content\/uploads\/2020\/01\/Integration-Rules-and-Formulas-3-256x300.jpg 256w\" sizes=\"(max-width: 500px) 100vw, 500px\" \/><br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-128714\" src=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2020\/01\/Integration-Rules-and-Formulas-4.jpg\" alt=\"Integration Rules and Formulas 4\" width=\"529\" height=\"714\" srcset=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2020\/01\/Integration-Rules-and-Formulas-4.jpg 529w, https:\/\/www.aplustopper.com\/wp-content\/uploads\/2020\/01\/Integration-Rules-and-Formulas-4-222x300.jpg 222w\" sizes=\"(max-width: 529px) 100vw, 529px\" \/><br \/>\nIn any of the fundamental integration formulae, if x is replaced by ax+b, then the same formulae is applicable but we must divide by coefficient of x or derivative of (ax+b) i.e., a. In general, if <strong>\u222b<\/strong>f(x) dx = \u03d5(x) + c, then<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-128715\" src=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2020\/01\/Integration-Rules-and-Formulas-5.jpg\" alt=\"Integration Rules and Formulas 5\" width=\"512\" height=\"172\" srcset=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2020\/01\/Integration-Rules-and-Formulas-5.jpg 512w, https:\/\/www.aplustopper.com\/wp-content\/uploads\/2020\/01\/Integration-Rules-and-Formulas-5-300x101.jpg 300w\" sizes=\"(max-width: 512px) 100vw, 512px\" \/><\/p>\n<h3>Some more Results<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-128718\" src=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2020\/01\/Integration-Rules-and-Formulas-7.jpg\" alt=\"Integration Rules and Formulas 7\" width=\"546\" height=\"681\" srcset=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2020\/01\/Integration-Rules-and-Formulas-7.jpg 546w, https:\/\/www.aplustopper.com\/wp-content\/uploads\/2020\/01\/Integration-Rules-and-Formulas-7-241x300.jpg 241w\" sizes=\"(max-width: 546px) 100vw, 546px\" \/><\/h3>\n","protected":false},"excerpt":{"rendered":"<p>Integration Rules and Formulas Integral of\u00a0a Function A function \u03d5(x) is called a primitive or an antiderivative of a function f(x), if ?'(x)\u00a0= f(x). Let f(x) be a function. Then the collection of all its primitives is called the indefinite integral of f(x) and is denoted by \u222bf(x) dx. Thus, where \u03d5(x) is primitive of [&hellip;]<\/p>\n","protected":false},"author":5,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_genesis_hide_title":false,"_genesis_hide_breadcrumbs":false,"_genesis_hide_singular_image":false,"_genesis_hide_footer_widgets":false,"_genesis_custom_body_class":"","_genesis_custom_post_class":"","_genesis_layout":"","footnotes":""},"categories":[5],"tags":[4740,4739,4738],"yoast_head":"<!-- This site is optimized with the Yoast SEO Premium plugin v17.8 (Yoast SEO v17.8) - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Integration Rules and Formulas - A Plus Topper<\/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:\/\/www.aplustopper.com\/integration-rules\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Integration Rules and Formulas\" \/>\n<meta property=\"og:description\" content=\"Integration Rules and Formulas Integral of\u00a0a Function A function \u03d5(x) is called a primitive or an antiderivative of a function f(x), if ?&#039;(x)\u00a0= f(x). Let f(x) be a function. Then the collection of all its primitives is called the indefinite integral of f(x) and is denoted by \u222bf(x) dx. Thus, where \u03d5(x) is primitive of [&hellip;]\" \/>\n<meta property=\"og:url\" content=\"https:\/\/www.aplustopper.com\/integration-rules\/\" \/>\n<meta property=\"og:site_name\" content=\"A Plus Topper\" \/>\n<meta property=\"article:publisher\" content=\"https:\/\/www.facebook.com\/aplustopper\/\" \/>\n<meta property=\"article:published_time\" content=\"2020-12-10T02:25:00+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2020-12-10T05:06:40+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2020\/01\/Integration-Rules-and-Formulas-1.jpg\" \/>\n<meta name=\"twitter:card\" content=\"summary\" \/>\n<meta name=\"twitter:label1\" content=\"Written by\" \/>\n\t<meta name=\"twitter:data1\" content=\"Prasanna\" \/>\n\t<meta name=\"twitter:label2\" content=\"Est. reading time\" \/>\n\t<meta name=\"twitter:data2\" content=\"1 minute\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"Organization\",\"@id\":\"https:\/\/www.aplustopper.com\/#organization\",\"name\":\"Aplus Topper\",\"url\":\"https:\/\/www.aplustopper.com\/\",\"sameAs\":[\"https:\/\/www.facebook.com\/aplustopper\/\"],\"logo\":{\"@type\":\"ImageObject\",\"@id\":\"https:\/\/www.aplustopper.com\/#logo\",\"inLanguage\":\"en-US\",\"url\":\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2018\/12\/Aplus_380x90-logo.jpg\",\"contentUrl\":\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2018\/12\/Aplus_380x90-logo.jpg\",\"width\":1585,\"height\":375,\"caption\":\"Aplus Topper\"},\"image\":{\"@id\":\"https:\/\/www.aplustopper.com\/#logo\"}},{\"@type\":\"WebSite\",\"@id\":\"https:\/\/www.aplustopper.com\/#website\",\"url\":\"https:\/\/www.aplustopper.com\/\",\"name\":\"A Plus Topper\",\"description\":\"Improve your Grades\",\"publisher\":{\"@id\":\"https:\/\/www.aplustopper.com\/#organization\"},\"potentialAction\":[{\"@type\":\"SearchAction\",\"target\":{\"@type\":\"EntryPoint\",\"urlTemplate\":\"https:\/\/www.aplustopper.com\/?s={search_term_string}\"},\"query-input\":\"required name=search_term_string\"}],\"inLanguage\":\"en-US\"},{\"@type\":\"ImageObject\",\"@id\":\"https:\/\/www.aplustopper.com\/integration-rules\/#primaryimage\",\"inLanguage\":\"en-US\",\"url\":\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2020\/01\/Integration-Rules-and-Formulas-1.jpg\",\"contentUrl\":\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2020\/01\/Integration-Rules-and-Formulas-1.jpg\",\"width\":364,\"height\":61,\"caption\":\"Integration Rules and Formulas 1\"},{\"@type\":\"WebPage\",\"@id\":\"https:\/\/www.aplustopper.com\/integration-rules\/#webpage\",\"url\":\"https:\/\/www.aplustopper.com\/integration-rules\/\",\"name\":\"Integration Rules and Formulas - 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Let f(x) be a function. Then the collection of all its primitives is called the indefinite integral of f(x) and is denoted by \u222bf(x) dx. Thus, where \u03d5(x) is primitive of [&hellip;]","og_url":"https:\/\/www.aplustopper.com\/integration-rules\/","og_site_name":"A Plus Topper","article_publisher":"https:\/\/www.facebook.com\/aplustopper\/","article_published_time":"2020-12-10T02:25:00+00:00","article_modified_time":"2020-12-10T05:06:40+00:00","og_image":[{"url":"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2020\/01\/Integration-Rules-and-Formulas-1.jpg"}],"twitter_card":"summary","twitter_misc":{"Written by":"Prasanna","Est. reading time":"1 minute"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"Organization","@id":"https:\/\/www.aplustopper.com\/#organization","name":"Aplus Topper","url":"https:\/\/www.aplustopper.com\/","sameAs":["https:\/\/www.facebook.com\/aplustopper\/"],"logo":{"@type":"ImageObject","@id":"https:\/\/www.aplustopper.com\/#logo","inLanguage":"en-US","url":"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2018\/12\/Aplus_380x90-logo.jpg","contentUrl":"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2018\/12\/Aplus_380x90-logo.jpg","width":1585,"height":375,"caption":"Aplus Topper"},"image":{"@id":"https:\/\/www.aplustopper.com\/#logo"}},{"@type":"WebSite","@id":"https:\/\/www.aplustopper.com\/#website","url":"https:\/\/www.aplustopper.com\/","name":"A Plus Topper","description":"Improve your Grades","publisher":{"@id":"https:\/\/www.aplustopper.com\/#organization"},"potentialAction":[{"@type":"SearchAction","target":{"@type":"EntryPoint","urlTemplate":"https:\/\/www.aplustopper.com\/?s={search_term_string}"},"query-input":"required name=search_term_string"}],"inLanguage":"en-US"},{"@type":"ImageObject","@id":"https:\/\/www.aplustopper.com\/integration-rules\/#primaryimage","inLanguage":"en-US","url":"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2020\/01\/Integration-Rules-and-Formulas-1.jpg","contentUrl":"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2020\/01\/Integration-Rules-and-Formulas-1.jpg","width":364,"height":61,"caption":"Integration Rules and Formulas 1"},{"@type":"WebPage","@id":"https:\/\/www.aplustopper.com\/integration-rules\/#webpage","url":"https:\/\/www.aplustopper.com\/integration-rules\/","name":"Integration Rules and Formulas - 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