{"id":87844,"date":"2021-08-10T14:24:35","date_gmt":"2021-08-10T19:24:35","guid":{"rendered":"https:\/\/www.mometrix.com\/academy\/?page_id=87844"},"modified":"2026-04-23T14:17:47","modified_gmt":"2026-04-23T19:17:47","slug":"integration-by-substitution","status":"publish","type":"page","link":"https:\/\/www.mometrix.com\/academy\/integration-by-substitution\/","title":{"rendered":"Integration by Substitution"},"content":{"rendered":"\n\t\t\t<div id=\"mmDeferVideoEncompass_wV4wgjMNhmo\" style=\"position: relative;\">\n\t\t\t<picture>\n\t\t\t\t<source srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/01\/circle-play-duotone.webp\" type=\"image\/webp\">\n\t\t\t\t<source srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/01\/circle-play-duotone.png\" type=\"image\/jpeg\"> \n\t\t\t\t<img fetchpriority=\"high\" decoding=\"async\" loading=\"eager\" id=\"videoThumbnailImage_wV4wgjMNhmo\" data-source-videoID=\"wV4wgjMNhmo\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/01\/circle-play-duotone.png\" alt=\"Integration by Substitution Video\" height=\"464\" width=\"825\" class=\"size-full\" data-matomo-title = \"Integration by Substitution\">\n\t\t\t<\/picture>\n\t\t\t<\/div>\n\t\t\t<style>img#videoThumbnailImage_wV4wgjMNhmo:hover {cursor:pointer;} img#videoThumbnailImage_wV4wgjMNhmo {background-size:contain;background-image:url(\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/02\/1766-thumb-final-1.webp\");}<\/style>\n\t\t\t<script defer>\n\t\t\t  jQuery(\"img#videoThumbnailImage_wV4wgjMNhmo\").click(function() {\n\t\t\t\tlet videoId = jQuery(this).attr(\"data-source-videoID\");\n\t\t\t\tlet helpTag = '<div id=\"mmDeferVideoYTMessage_wV4wgjMNhmo\" style=\"display: none;position: absolute;top: -24px;width: 100%;text-align: center;\"><span style=\"font-style: italic;font-size: small;border-top: 1px solid #fc0;\">Having trouble? <a href=\"https:\/\/www.youtube.com\/watch?v='+videoId+'\" target=\"_blank\">Click here to watch on YouTube.<\/a><\/span><\/div>';\n\t\t\t\tlet tag = document.createElement(\"iframe\");\n\t\t\t\ttag.id = \"yt\" + videoId;\n\t\t\t\ttag.src = \"https:\/\/www.youtube-nocookie.com\/embed\/\" + videoId + \"?autoplay=1&controls=1&wmode=opaque&rel=0&egm=0&iv_load_policy=3&hd=0&enablejsapi=1\";\n\t\t\t\ttag.frameborder = 0;\n\t\t\t\ttag.allow = \"autoplay; fullscreen\";\n\t\t\t\ttag.width = this.width;\n\t\t\t\ttag.height = this.height;\n\t\t\t\ttag.setAttribute(\"data-matomo-title\",\"Integration by Substitution\");\n\t\t\t\tjQuery(\"div#mmDeferVideoEncompass_wV4wgjMNhmo\").html(tag);\n\t\t\t\tjQuery(\"div#mmDeferVideoEncompass_wV4wgjMNhmo\").prepend(helpTag);\n\t\t\t\tsetTimeout(function(){jQuery(\"div#mmDeferVideoYTMessage_wV4wgjMNhmo\").css(\"display\", \"block\");}, 2000);\n\t\t\t  });\n\t\t\t  \n\t\t\t<\/script>\n\t\t\n<p><script>\nfunction SMg_Function() {\n  var x = document.getElementById(\"SMg\");\n  if (x.style.display === \"none\") {\n    x.style.display = \"block\";\n  } else {\n    x.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<div class=\"moc-toc hide-on-desktop hide-on-tablet\">\n<div><button onclick=\"SMg_Function()\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2024\/12\/toc2.svg\" width=\"16\" height=\"16\" alt=\"show or hide table of contents\"><\/button><\/p>\n<p>On this page<\/p>\n<\/div>\n<nav id=\"SMg\" style=\"display:none;\">\n<ul>\n<li class=\"toc-h2\"><a href=\"#Integration_and_Differentiation_as_Inverse_Operations\" class=\"smooth-scroll\">Integration and Differentiation as Inverse Operations<\/a><\/li>\n<li class=\"toc-h2\"><a href=\"#How_USubstitution_Works\" class=\"smooth-scroll\">How U-Substitution Works<\/a><\/li>\n<li class=\"toc-h2\"><a href=\"#Practice_Problem\" class=\"smooth-scroll\">Practice Problem<\/a><\/li>\n<\/ul>\n<\/nav>\n<\/div>\n<div class=\"accordion\"><input id=\"transcript\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"transcript\">Transcript<\/label>\n<div class=\"spoiler\" id=\"transcript-spoiler\">\n<h2><span id=\"Integration_and_Differentiation_as_Inverse_Operations\" class=\"m-toc-anchor\"><\/span>Integration and Differentiation as Inverse Operations<\/h2>\n<p>\nAccording to the fundamental theorem of calculus, integration and differentiation can be thought of as inverse operations. Differentiating an integral and integrating a derivative yield almost identical results. For example:<\/p>\n<div class=\"examplesentence\">\n\\(\\large{\\frac{d}{dx}}\\normalsize{\\left [ \\int 3x^{2}dx \\right ]=3x^{2}}\\)<br \/>\n\\(\\int (\\large{\\frac{d}{dx}}\\normalsize{\\left [ 2x^{4}+7 \\right ])dx=2x^{4}+c}\\)\n<\/div>\n<p>\n&nbsp;<br \/>\nFunction types such as polynomial functions, exponential functions, and trigonometric functions have their own differentiation and integration formulas. When it comes to techniques, however, integration and differentiation can be quite different. In general, when taking a derivative, if the function is a polynomial, use the power rule, if it is a product, use the product rule, if it is a quotient, use the quotient rule, and if it is a composite, use the chain rule. <\/p>\n<p>It doesn\u2019t work that way with integration. There\u2019s no rule specifically for integrating a product or a quotient. Part of the reason is that the product, quotient, and chain rules all incorporate parts of the function being differentiated as well as the derivatives of those parts. So, when looking at an integral, it\u2019s difficult to tell which rule needs to be undone.<\/p>\n<h2><span id=\"How_USubstitution_Works\" class=\"m-toc-anchor\"><\/span>How U-Substitution Works<\/h2>\n<p>\nFortunately, there is a useful technique that involves substitution.<\/p>\n<h3><span id=\"Example_1\" class=\"m-toc-anchor\"><\/span>Example 1<\/h3>\n<p>\nLet\u2019s begin with an example:<\/p>\n<div class=\"examplesentence\">\\(\\int (8x-13)(4x^{2}-13x)^{5}dx\\)<\/div>\n<p>\n&nbsp;<br \/>\nAt first glance, this seems like a quick polynomial integration, right? Look closely. In order to integrate this as a polynomial, we would need to expand <span style=\"font-style:normal; font-size:90%\">\\(4x^{2}-13x\\)<\/span> to the 5th power, then distribute the <span style=\"font-style:normal; font-size:90%\">\\(8x-13\\)<\/span> to that gigantic result (a huge amount of work)!<\/p>\n<p>Notice that <span style=\"font-style:normal; font-size:90%\">\\(8x-13\\)<\/span> is the derivative of <span style=\"font-style:normal; font-size:90%\">\\(4x^{2}-13x\\)<\/span>. This tells us we can use substitution. Traditionally, the variable <span style=\"font-style:normal; font-size:90%\">\\(u\\)<\/span> is used in this technique, so it is often called <span style=\"font-style:normal; font-size:90%\">\\(u\\)<\/span>-substitution.<\/p>\n<p>So here\u2019s the big idea:<\/p>\n<ol>\n<li>First, we\u2019re gonna assign a part of the function to be <span style=\"font-style:normal; font-size:90%\">\\(u\\)<\/span>.<\/li>\n<li>Then we\u2019ll rewrite the integral in terms of <span style=\"font-style:normal; font-size:90%\">\\(u\\)<\/span>, not <span style=\"font-style:normal; font-size:90%\">\\(x\\)<\/span>, creating an easily-integrable form.<\/li>\n<li>Then we\u2019ll integrate.<\/li>\n<li>Finally, we\u2019ll convert the result back to being in terms of <span style=\"font-style:normal; font-size:90%\">\\(x\\)<\/span>.<\/li>\n<\/ol>\n<h4 style=\"margin-bottom: 0em; font-weight: 600 !important; font-size: 105%\"><span id=\"Choose\" class=\"m-toc-anchor\"><\/span>Choose \\(u\\)<\/h4>\n<p>\nFirst, identify the part of the function that is the derivative (or multiple of the derivative) of the other. Once this is identified, the other piece will be <span style=\"font-style:normal; font-size:90%\">\\(u\\)<\/span>. Typically, this involves thinking a little bit ahead and making a choice. In this case, since <span style=\"font-style:normal; font-size:90%\">\\(8x-13\\)<\/span> is the derivative of <span style=\"font-style:normal; font-size:90%\">\\(4x^{2}-13x\\)<\/span>, we\u2019ll call <span style=\"font-style:normal; font-size:90%\">\\(4x^{2}-13x\\)<\/span> our <span style=\"font-style:normal; font-size:90%\">\\(u\\)<\/span>.<\/p>\n<p>So <span style=\"font-style:normal; font-size:90%\">\\(u=4x^{2}-13x\\)<\/span>.<\/p>\n<h4 style=\"margin-bottom: 0em; font-weight: 600 !important; font-size: 105%\"><span id=\"Find\" class=\"m-toc-anchor\"><\/span>Find \\(du\\)<\/h4>\n<p>\nThen we\u2019ll take the derivative of this, and get <span style=\"font-style:normal; font-size:90%\">\\(du=(8x-13)dx\\)<\/span>.<\/p>\n<p>This gives us everything we need to rewrite the integral in terms of <span style=\"font-style:normal; font-size:90%\">\\(u\\)<\/span>.<\/p>\n<h4 style=\"margin-bottom: 0em; font-weight: 600 !important; font-size: 105%\"><span id=\"Rewrite_the_Integral\" class=\"m-toc-anchor\"><\/span>Rewrite the Integral<\/h4>\n<p>\nSo we\u2019re gonna have our <span style=\"font-style:normal; font-size:90%\">\\(u\\)<\/span> is right here, this part <span style=\"font-style:normal; font-size:90%\">\\((4x^{2}-13x)\\)<\/span>, not including to the 5th power. And then our <span style=\"font-style:normal; font-size:90%\">\\(du\\)<\/span> is this part <span style=\"font-style:normal; font-size:90%\">\\((8x-13)\\)<\/span> and this part <span style=\"font-style:normal; font-size:90%\">\\((dx)\\)<\/span>.<\/p>\n<p>So if we rewrite this integral, we get:<\/p>\n<div class=\"examplesentence\">\\(\\int u^{5}du\\)<\/div>\n<p>\n&nbsp;<\/p>\n<h4 style=\"margin-bottom: 0em; font-weight: 600 !important; font-size: 105%\"><span id=\"Integrate\" class=\"m-toc-anchor\"><\/span>Integrate<\/h4>\n<p>\nNow this becomes something that can be easily integrated. So this is equal to:<\/p>\n<div class=\"examplesentence\">\\(\\int u^{5}du=\\large{\\frac{1}{6}}\\normalsize{u^{6}+c}\\)<\/div>\n<p>\n&nbsp;<\/p>\n<h4 style=\"margin-bottom: 0em; font-weight: 600 !important; font-size: 105%\"><span id=\"Substitute\" class=\"m-toc-anchor\"><\/span>Substitute<\/h4>\n<p>\nAll that\u2019s left is to rewrite the result in terms of <span style=\"font-style:normal; font-size:90%\">\\(x\\)<\/span>, which involves substituting <span style=\"font-style:normal; font-size:90%\">\\(4x^{2}-13x\\)<\/span> for <span style=\"font-style:normal; font-size:90%\">\\(u\\)<\/span>. So we have this being equal to:<\/p>\n<div class=\"examplesentence\">\\(\\large{\\frac{1}{6}}\\normalsize{(4x^{2}-13x)^{6}+c}\\)<\/div>\n<p>\n&nbsp;<br \/>\nRemember, you can always take the derivative to check your result.<\/p>\n<h3><span id=\"Example_2\" class=\"m-toc-anchor\"><\/span>Example 2<\/h3>\n<p>\nLet\u2019s try another!<\/p>\n<div class=\"examplesentence\">\\(\\int 6(x-4)\\sqrt[\\large{3}]{x^{2}-8x}dx\\)<\/div>\n<p>\n&nbsp;<br \/>\nAt first glance, there are many possibilities for <span style=\"font-style:normal; font-size:90%\">\\(u\\)<\/span>: <span style=\"font-style:normal; font-size:90%\">\\(x-4\\)<\/span>, <span style=\"font-style:normal; font-size:90%\">\\((x-4)^{3}\\)<\/span>, <span style=\"font-style:normal; font-size:90%\">\\(x^{2}-8x\\)<\/span>, <span style=\"font-style:normal; font-size:90%\">\\((x^{2}-8x)^{\\frac{1}{3}}\\)<\/span>, but there is only one that has the necessary relationship.<\/p>\n<h4 style=\"margin-bottom: 0em; font-weight: 600 !important; font-size: 105%\"><span id=\"Choose\" class=\"m-toc-anchor\"><\/span>Choose \\(u\\)<\/h4>\n<p>\nLet\u2019s take a look at:<\/p>\n<div class=\"examplesentence\">\\(\\large{\\frac{d}{dx}}\\normalsize{\\left [ x^{2}-8x \\right ]}\\)<\/div>\n<p>\n&nbsp;<br \/>\nThis gives us:<\/p>\n<div class=\"examplesentence\">\\(\\large{\\frac{d}{dx}}\\normalsize{\\left [ x^{2}-8x \\right ]=2x-8}\\)<\/div>\n<p>\n&nbsp;<br \/>\nWhich then can factor to be:<\/p>\n<div class=\"examplesentence\">\\(\\large{\\frac{d}{dx}}\\normalsize{\\left [ x^{2}-8x \\right ]=2x-8}\\)\\(\\:=2(x-4)\\)<\/div>\n<p>\n&nbsp;<br \/>\nSince <span style=\"font-style:normal; font-size:90%\">\\(x-4\\)<\/span> is a multiple of the derivative of <span style=\"font-style:normal; font-size:90%\">\\(x^{2}-8x\\)<\/span>, <span style=\"font-style:normal; font-size:90%\">\\(x^{2}-8x\\)<\/span> must be our <span style=\"font-style:normal; font-size:90%\">\\(u\\)<\/span>.<\/p>\n<p>So <span style=\"font-style:normal; font-size:90%\">\\(u=x^{2}-8x\\)<\/span>, and <span style=\"font-style:normal; font-size:90%\">\\(du=(2x-8)dx\\)<\/span>, which is <span style=\"font-style:normal; font-size:90%\">\\(2(x-4)dx\\)<\/span>.<\/p>\n<h4 style=\"margin-bottom: 0em; font-weight: 600 !important; font-size: 105%\"><span id=\"Find\" class=\"m-toc-anchor\"><\/span>Find \\(du\\)<\/h4>\n<p>\nNotice that there\u2019s no 2 in the original integral. No problem, just divide by 2:<\/p>\n<div class=\"examplesentence\">\\(\\large{\\frac{1}{2}}\\normalsize{du=(x-4)dx}\\)<\/div>\n<p>\n&nbsp;<\/p>\n<h4 style=\"margin-bottom: 0em; font-weight: 600 !important; font-size: 105%\"><span id=\"Rewrite_the_Integral\" class=\"m-toc-anchor\"><\/span>Rewrite the Integral<\/h4>\n<p>\nMake the substitutions and simplify a little bit.<\/p>\n<p>So notice, up here we have our <span style=\"font-style:normal; font-size:90%\">\\(u\\)<\/span> is, <span style=\"font-style:normal; font-size:90%\">\\(x^{2}-8x\\)<\/span>, so that\u2019s this portion. And then our <span style=\"font-style:normal; font-size:90%\">\\(du\\)<\/span>, or <span style=\"font-style:normal; font-size:90%\">\\(\\frac{1}{2}du\\)<\/span>, is <span style=\"font-style:normal; font-size:90%\">\\((x-4)dx\\)<\/span>. Now let\u2019s substitute in our <span style=\"font-style:normal; font-size:90%\">\\(u\\)<\/span>\u2019s.<\/p>\n<p>The integral of 6, <span style=\"font-style:normal; font-size:90%\">\\(u=x^{2}-8x\\)<\/span>, which is this part, cube rooted, or we can say to the <span style=\"font-style:normal; font-size:90%\">\\(\\frac{1}{3}\\)<\/span> power (it means the same thing, times our blue part right here, which we said is <span style=\"font-style:normal; font-size:90%\">\\(\\frac{1}{2}du\\)<\/span>).<\/p>\n<div class=\"examplesentence\">\\(\\int 6\\cdot u^{\\large{\\frac{1}{3}}}\\normalsize{\\cdot} \\large{\\frac{1}{2}}\\normalsize{du}\\)<\/div>\n<p>\n&nbsp;<\/p>\n<p>When we simplify this, we get:<\/p>\n<div class=\"examplesentence\">\\(\\int 6\\cdot u^{\\large{\\frac{1}{3}}}\\normalsize{\\cdot} \\large{\\frac{1}{2}}\\normalsize{du}\\)\\(\\:=\\int 3u^{\\large{\\frac{1}{3}}}\\normalsize{du}\\)<\/div>\n<p>\n&nbsp;<\/p>\n<h4 style=\"margin-bottom: 0em; font-weight: 600 !important; font-size: 105%\"><span id=\"Integrate\" class=\"m-toc-anchor\"><\/span>Integrate<\/h4>\n<p>\nWhich is equal to:<\/p>\n<div class=\"examplesentence\">\\(\\large{\\frac{3}{4}}\\normalsize{\\cdot 3u}^{\\large{\\frac{4}{3}}}\\normalsize{+c=}\\large{\\frac{9}{4}}\\normalsize{u}^{\\large{\\frac{4}{3}}}\\normalsize{+c}\\)<\/div>\n<p>\n&nbsp;<\/p>\n<h4 style=\"margin-bottom: 0em; font-weight: 600 !important; font-size: 105%\"><span id=\"Substitute_Back\" class=\"m-toc-anchor\"><\/span>Substitute Back<\/h4>\n<p>\nNow we\u2019re gonna rewrite this in terms of <span style=\"font-style:normal; font-size:90%\">\\(x\\)<\/span>. <\/p>\n<div class=\"examplesentence\">\\(\\int 6(x-4)\\sqrt[\\large{3}]{x^{2}-8x}dx\\)\\(\\:=\\large{\\frac{9}{4}}\\normalsize{(x^{2}-8x)}^{\\large{\\frac{4}{3}}}\\normalsize{+C}\\)<\/div>\n<p>\n&nbsp;<\/p>\n<h3><span id=\"Example_3\" class=\"m-toc-anchor\"><\/span>Example 3<\/h3>\n<p>\nLet\u2019s work one final problem together.<\/p>\n<div class=\"examplesentence\">\\(\\int \\cos(9x)dx\\)<\/div>\n<p>\n&nbsp;<br \/>\nThe temptation here is to say that the answer is <span style=\"font-style:normal; font-size:90%\">\\(\\sin(9x)+c\\)<\/span> since we know the derivative of <span style=\"font-style:normal; font-size:90%\">\\(\\sin(x)\\)<\/span> is <span style=\"font-style:normal; font-size:90%\">\\(\\cos(x)\\)<\/span>. But we need to account for the \\(9x\\) in the parentheses since it is a function.<\/p>\n<p>Clearly, we can\u2019t make <span style=\"font-style:normal; font-size:90%\">\\(u\\)<\/span> equal to <span style=\"font-style:normal; font-size:90%\">\\(\\cos(9x)\\)<\/span> because there is no negative sine function in the integral. So if our <span style=\"font-style:normal; font-size:90%\">\\(u=9x\\)<\/span>, then <span style=\"font-style:normal; font-size:90%\">\\(du\\)<\/span> must be equal to <span style=\"font-style:normal; font-size:90%\">\\(9dx\\)<\/span>. But there\u2019s not a 9 in our function, so we\u2019re gonna take <span style=\"font-style:normal; font-size:90%\">\\(\\frac{1}{9}du=dx\\)<\/span>.<\/p>\n<p>Now let\u2019s work on rewriting our integral:<\/p>\n<div class=\"examplesentence\">\\(\\int \\large{\\frac{1}{9}}\\normalsize{\\cos(u)du}\\)<\/div>\n<p>\n&nbsp;<br \/>\nNow we\u2019re going to integrate. So this is equal to:<\/p>\n<div class=\"examplesentence\">\\(\\large{\\frac{1}{9}}\\normalsize{\\sin(u)+c}\\)<\/div>\n<p>\n&nbsp;<br \/>\nAnd finally, we\u2019re going to substitute in our <span style=\"font-style:normal; font-size:90%\">\\(u\\)<\/span>-value again. So this gives us:<\/p>\n<div class=\"examplesentence\">\\(\\large{\\frac{1}{9}}\\normalsize{\\sin(9x)+c}\\)<\/div>\n<p>\n&nbsp;<\/p>\n<h2><span id=\"Practice_Problem\" class=\"m-toc-anchor\"><\/span>Practice Problem<\/h2>\n<p>\nHere\u2019s one for you to try! Using substitution, find a formula for the antiderivative of the tangent function.<\/p>\n<div class=\"examplesentence\">\\(\\int \\tan(x)dx=\\int \\large{\\frac{\\sin(x)}{\\cos(x)}}\\normalsize{dx}\\)<\/div>\n<p>\n&nbsp;<br \/>\nPause the video and try this one for yourself! Okay, so the first thing we need to do is choose our <span style=\"font-style:normal; font-size:90%\">\\(u\\)<\/span> carefully, because <span style=\"font-style:normal; font-size:90%\">\\(du\\)<\/span> cannot be in a denominator.<\/p>\n<p>So our <span style=\"font-style:normal; font-size:90%\">\\(u\\)<\/span> is gonna be <span style=\"font-style:normal; font-size:90%\">\\(\\cos(x)\\)<\/span>. This then means that our <span style=\"font-style:normal; font-size:90%\">\\(du=-\\sin(x)dx\\)<\/span>. Now, there\u2019s not a <span style=\"font-style:normal; font-size:90%\">\\(-\\sin(x)\\)<\/span> in our function, so we\u2019re gonna make this <span style=\"font-style:normal; font-size:90%\">\\(-du\\)<\/span>, by dividing both sides by negative 1, <span style=\"font-style:normal; font-size:90%\">\\(-du=\\sin(x)dx\\)<\/span>.<\/p>\n<p>Now let\u2019s rewrite our function. When we rewrite this, we\u2019ll have:<\/p>\n<div class=\"examplesentence\">\\(\\int -\\large{\\frac{1}{u}}\\normalsize{du}\\)<\/div>\n<p>\n&nbsp;<br \/>\nWhen we integrate this, we get:<\/p>\n<div class=\"examplesentence\">\\(-\\ln\\left | u \\right |+c\\)<\/div>\n<p>\n&nbsp;<br \/>\nFinally, we\u2019re gonna rewrite this in terms of <span style=\"font-style:normal; font-size:90%\">\\(x\\)<\/span>. So this is equal to:<\/p>\n<div class=\"examplesentence\">\\(-\\ln\\left | \\cos(x) \\right |+c\\)<\/div>\n<p>\n&nbsp;<br \/>\nBe sure to integrate the method of substitution into your calculus repertoire! It is one of the key ways of finding antiderivatives.<\/p>\n<p>I hope you enjoyed this video! Thanks for watching and happy studying!<\/p>\n<\/div>\n<\/div>\n\n<div class=\"home-buttons\">\n<p><a href=\"https:\/\/www.mometrix.com\/academy\/calculus\/\">Return to Calculus Videos<\/a><\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>Return to Calculus Videos<\/p>\n","protected":false},"author":1,"featured_media":100777,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":{"0":"post-87844","1":"page","2":"type-page","3":"status-publish","4":"has-post-thumbnail","6":"page_category-calculus-videos","7":"page_category-video-pages-for-study-course-sidebar-ad","8":"page_type-video","9":"subject_matter-math"},"aioseo_notices":[],"aioseo_head":"\n\t\t<!-- All in One SEO Pro 4.9.8 - aioseo.com -->\n\t<meta name=\"description\" content=\"Integration and differentiation are considered inverse functions in calculus. Learn how to integrate by substitution with examples in this video!\" \/>\n\t<meta name=\"robots\" content=\"max-image-preview:large\" \/>\n\t<link rel=\"canonical\" href=\"https:\/\/www.mometrix.com\/academy\/integration-by-substitution\/\" \/>\n\t<meta name=\"generator\" content=\"All in One SEO Pro (AIOSEO) 4.9.8\" \/>\n\t\t<meta property=\"og:locale\" content=\"en_US\" \/>\n\t\t<meta property=\"og:site_name\" content=\"|\" \/>\n\t\t<meta property=\"og:type\" content=\"article\" \/>\n\t\t<meta property=\"og:title\" content=\"Integration by Substitution (Video)\" \/>\n\t\t<meta property=\"og:description\" content=\"Integration and differentiation are considered inverse functions in calculus. Learn how to integrate by substitution with examples in this video!\" \/>\n\t\t<meta property=\"og:url\" content=\"https:\/\/www.mometrix.com\/academy\/integration-by-substitution\/\" \/>\n\t\t<meta property=\"og:image\" content=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/11\/1766-thumb-final.jpg\" \/>\n\t\t<meta property=\"og:image:secure_url\" content=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/11\/1766-thumb-final.jpg\" \/>\n\t\t<meta property=\"og:image:width\" content=\"1920\" \/>\n\t\t<meta property=\"og:image:height\" content=\"1080\" \/>\n\t\t<meta property=\"article:published_time\" content=\"2021-08-10T19:24:35+00:00\" \/>\n\t\t<meta property=\"article:modified_time\" content=\"2026-04-23T19:17:47+00:00\" \/>\n\t\t<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n\t\t<meta name=\"twitter:title\" content=\"Integration by Substitution (Video)\" \/>\n\t\t<meta name=\"twitter:description\" content=\"Integration and differentiation are considered inverse functions in calculus. Learn how to integrate by substitution with examples in this video!\" \/>\n\t\t<meta name=\"twitter:image\" content=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/12\/Mometrix-Academy-FI.png\" \/>\n\t\t<script type=\"application\/ld+json\" class=\"aioseo-schema\">\n\t\t\t{\"@context\":\"https:\\\/\\\/schema.org\",\"@graph\":[{\"@type\":\"BreadcrumbList\",\"@id\":\"https:\\\/\\\/www.mometrix.com\\\/academy\\\/integration-by-substitution\\\/#breadcrumblist\",\"itemListElement\":[{\"@type\":\"ListItem\",\"@id\":\"https:\\\/\\\/www.mometrix.com\\\/academy#listItem\",\"position\":1,\"name\":\"Home\",\"item\":\"https:\\\/\\\/www.mometrix.com\\\/academy\",\"nextItem\":{\"@type\":\"ListItem\",\"@id\":\"https:\\\/\\\/www.mometrix.com\\\/academy\\\/integration-by-substitution\\\/#listItem\",\"name\":\"Integration by Substitution\"}},{\"@type\":\"ListItem\",\"@id\":\"https:\\\/\\\/www.mometrix.com\\\/academy\\\/integration-by-substitution\\\/#listItem\",\"position\":2,\"name\":\"Integration by Substitution\",\"previousItem\":{\"@type\":\"ListItem\",\"@id\":\"https:\\\/\\\/www.mometrix.com\\\/academy#listItem\",\"name\":\"Home\"}}]},{\"@type\":\"Organization\",\"@id\":\"https:\\\/\\\/www.mometrix.com\\\/academy\\\/#organization\",\"url\":\"https:\\\/\\\/www.mometrix.com\\\/academy\\\/\",\"logo\":{\"@type\":\"ImageObject\",\"url\":\"https:\\\/\\\/www.mometrix.com\\\/academy\\\/wp-content\\\/uploads\\\/2022\\\/06\\\/Mometrix-Test-Prep-Logo-min.png\",\"@id\":\"https:\\\/\\\/www.mometrix.com\\\/academy\\\/integration-by-substitution\\\/#organizationLogo\",\"width\":557,\"height\":242,\"caption\":\"Mometrix Test Preparation logo\"},\"image\":{\"@id\":\"https:\\\/\\\/www.mometrix.com\\\/academy\\\/integration-by-substitution\\\/#organizationLogo\"}},{\"@type\":\"WebPage\",\"@id\":\"https:\\\/\\\/www.mometrix.com\\\/academy\\\/integration-by-substitution\\\/#webpage\",\"url\":\"https:\\\/\\\/www.mometrix.com\\\/academy\\\/integration-by-substitution\\\/\",\"name\":\"Integration by Substitution (Video)\",\"description\":\"Integration and differentiation are considered inverse functions in calculus. Learn how to integrate by substitution with examples in this video!\",\"inLanguage\":\"en-US\",\"isPartOf\":{\"@id\":\"https:\\\/\\\/www.mometrix.com\\\/academy\\\/#website\"},\"breadcrumb\":{\"@id\":\"https:\\\/\\\/www.mometrix.com\\\/academy\\\/integration-by-substitution\\\/#breadcrumblist\"},\"image\":{\"@type\":\"ImageObject\",\"url\":\"https:\\\/\\\/www.mometrix.com\\\/academy\\\/wp-content\\\/uploads\\\/2021\\\/11\\\/1766-thumb-final.jpg\",\"@id\":\"https:\\\/\\\/www.mometrix.com\\\/academy\\\/integration-by-substitution\\\/#mainImage\",\"width\":1920,\"height\":1080,\"caption\":\"Person in an orange shirt standing against a gray background. Text reads, \\\"Integration by Substitution\\\" and \\\"Mometrix Test Preparation.\"},\"primaryImageOfPage\":{\"@id\":\"https:\\\/\\\/www.mometrix.com\\\/academy\\\/integration-by-substitution\\\/#mainImage\"},\"datePublished\":\"2021-08-10T14:24:35-05:00\",\"dateModified\":\"2026-04-23T14:17:47-05:00\"},{\"@type\":\"WebSite\",\"@id\":\"https:\\\/\\\/www.mometrix.com\\\/academy\\\/#website\",\"url\":\"https:\\\/\\\/www.mometrix.com\\\/academy\\\/\",\"inLanguage\":\"en-US\",\"publisher\":{\"@id\":\"https:\\\/\\\/www.mometrix.com\\\/academy\\\/#organization\"}}]}\n\t\t<\/script>\n\t\t<!-- All in One SEO Pro -->\r\n\t\t<title>Integration by Substitution (Video)<\/title>\n\n","aioseo_head_json":{"title":"Integration by Substitution (Video)","description":"Integration and differentiation are considered inverse functions in calculus. Learn how to integrate by substitution with examples in this video!","canonical_url":"https:\/\/www.mometrix.com\/academy\/integration-by-substitution\/","robots":"max-image-preview:large","keywords":"","webmasterTools":{"miscellaneous":""},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"BreadcrumbList","@id":"https:\/\/www.mometrix.com\/academy\/integration-by-substitution\/#breadcrumblist","itemListElement":[{"@type":"ListItem","@id":"https:\/\/www.mometrix.com\/academy#listItem","position":1,"name":"Home","item":"https:\/\/www.mometrix.com\/academy","nextItem":{"@type":"ListItem","@id":"https:\/\/www.mometrix.com\/academy\/integration-by-substitution\/#listItem","name":"Integration by Substitution"}},{"@type":"ListItem","@id":"https:\/\/www.mometrix.com\/academy\/integration-by-substitution\/#listItem","position":2,"name":"Integration by Substitution","previousItem":{"@type":"ListItem","@id":"https:\/\/www.mometrix.com\/academy#listItem","name":"Home"}}]},{"@type":"Organization","@id":"https:\/\/www.mometrix.com\/academy\/#organization","url":"https:\/\/www.mometrix.com\/academy\/","logo":{"@type":"ImageObject","url":"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2022\/06\/Mometrix-Test-Prep-Logo-min.png","@id":"https:\/\/www.mometrix.com\/academy\/integration-by-substitution\/#organizationLogo","width":557,"height":242,"caption":"Mometrix Test Preparation logo"},"image":{"@id":"https:\/\/www.mometrix.com\/academy\/integration-by-substitution\/#organizationLogo"}},{"@type":"WebPage","@id":"https:\/\/www.mometrix.com\/academy\/integration-by-substitution\/#webpage","url":"https:\/\/www.mometrix.com\/academy\/integration-by-substitution\/","name":"Integration by Substitution (Video)","description":"Integration and differentiation are considered inverse functions in calculus. Learn how to integrate by substitution with examples in this video!","inLanguage":"en-US","isPartOf":{"@id":"https:\/\/www.mometrix.com\/academy\/#website"},"breadcrumb":{"@id":"https:\/\/www.mometrix.com\/academy\/integration-by-substitution\/#breadcrumblist"},"image":{"@type":"ImageObject","url":"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/11\/1766-thumb-final.jpg","@id":"https:\/\/www.mometrix.com\/academy\/integration-by-substitution\/#mainImage","width":1920,"height":1080,"caption":"Person in an orange shirt standing against a gray background. Text reads, \"Integration by Substitution\" and \"Mometrix Test Preparation."},"primaryImageOfPage":{"@id":"https:\/\/www.mometrix.com\/academy\/integration-by-substitution\/#mainImage"},"datePublished":"2021-08-10T14:24:35-05:00","dateModified":"2026-04-23T14:17:47-05:00"},{"@type":"WebSite","@id":"https:\/\/www.mometrix.com\/academy\/#website","url":"https:\/\/www.mometrix.com\/academy\/","inLanguage":"en-US","publisher":{"@id":"https:\/\/www.mometrix.com\/academy\/#organization"}}]},"og:locale":"en_US","og:site_name":"|","og:type":"article","og:title":"Integration by Substitution (Video)","og:description":"Integration and differentiation are considered inverse functions in calculus. Learn how to integrate by substitution with examples in this video!","og:url":"https:\/\/www.mometrix.com\/academy\/integration-by-substitution\/","og:image":"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/11\/1766-thumb-final.jpg","og:image:secure_url":"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/11\/1766-thumb-final.jpg","og:image:width":"1920","og:image:height":"1080","article:published_time":"2021-08-10T19:24:35+00:00","article:modified_time":"2026-04-23T19:17:47+00:00","twitter:card":"summary_large_image","twitter:title":"Integration by Substitution (Video)","twitter:description":"Integration and differentiation are considered inverse functions in calculus. Learn how to integrate by substitution with examples in this video!","twitter:image":"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/12\/Mometrix-Academy-FI.png"},"aioseo_meta_data":{"post_id":"87844","title":"Integration by Substitution (Video)","description":"Integration and differentiation are considered inverse functions in calculus. Learn how to integrate by substitution with examples in this video!","keywords":null,"keyphrases":{"focus":{"keyphrase":"","score":0,"analysis":{"keyphraseInTitle":{"score":0,"maxScore":9,"error":1}}},"additional":[]},"primary_term":null,"canonical_url":null,"og_title":null,"og_description":null,"og_object_type":"default","og_image_type":"featured","og_image_url":"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/11\/1766-thumb-final.jpg","og_image_width":"1920","og_image_height":"1080","og_image_custom_url":null,"og_image_custom_fields":null,"og_video":"","og_custom_url":null,"og_article_section":null,"og_article_tags":null,"twitter_use_og":false,"twitter_card":"default","twitter_image_type":"default","twitter_image_url":null,"twitter_image_custom_url":null,"twitter_image_custom_fields":null,"twitter_title":null,"twitter_description":null,"schema":{"blockGraphs":[],"customGraphs":[],"default":{"data":{"Article":[],"Course":[],"Dataset":[],"FAQPage":[],"Movie":[],"Person":[],"Product":[],"ProductReview":[],"Car":[],"Recipe":[],"Service":[],"SoftwareApplication":[],"WebPage":[]},"graphName":"WebPage","isEnabled":true},"graphs":[]},"schema_type":"default","schema_type_options":"{\"article\":{\"articleType\":\"BlogPosting\"},\"course\":{\"name\":\"\",\"description\":\"\",\"provider\":\"\"},\"faq\":{\"pages\":[]},\"product\":{\"reviews\":[]},\"recipe\":{\"ingredients\":[],\"instructions\":[],\"keywords\":[]},\"software\":{\"reviews\":[],\"operatingSystems\":[]},\"webPage\":{\"webPageType\":\"WebPage\"}}","pillar_content":false,"robots_default":true,"robots_noindex":false,"robots_noarchive":false,"robots_nosnippet":false,"robots_nofollow":false,"robots_noimageindex":false,"robots_noodp":false,"robots_notranslate":false,"robots_max_snippet":"-1","robots_max_videopreview":"-1","robots_max_imagepreview":"large","priority":null,"frequency":"default","location":null,"local_seo":null,"seo_analyzer_scan_date":null,"breadcrumb_settings":null,"limit_modified_date":false,"open_ai":"{\"title\":{\"suggestions\":[],\"usage\":0},\"description\":{\"suggestions\":[],\"usage\":0}}","ai":{"faqs":[],"keyPoints":[],"schemas":[],"titles":[],"descriptions":[],"socialPosts":{"email":[],"linkedin":[],"twitter":[],"facebook":[],"instagram":[]}},"created":"2021-08-10 18:13:32","updated":"2026-04-23 19:59:09"},"_links":{"self":[{"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/pages\/87844","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/comments?post=87844"}],"version-history":[{"count":5,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/pages\/87844\/revisions"}],"predecessor-version":[{"id":292190,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/pages\/87844\/revisions\/292190"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/media\/100777"}],"wp:attachment":[{"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/media?parent=87844"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}