{"id":232071,"date":"2025-06-11T16:44:54","date_gmt":"2025-06-11T16:44:54","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=232071"},"modified":"2025-06-11T16:44:56","modified_gmt":"2025-06-11T16:44:56","slug":"consider-the-following-integral-2","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/06\/11\/consider-the-following-integral-2\/","title":{"rendered":"Consider the following integral."},"content":{"rendered":"\n<p>Consider the following integral.<br>Find a substitution to rewrite the integrand as<\/p>\n\n\n\n<p>.<\/p>\n\n\n\n<p>Evaluate the given integral. (Use C for the constant of integration. Remember to use absolute values where appropriate.)<\/p>\n\n\n\n<p><strong><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-1-color\">The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n<p><strong>Answer:<\/strong><br>The required substitution is&nbsp;<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>u=ex+6<em>u<\/em>=<em>e<\/em><em>x<\/em>+6<\/code><\/pre>\n\n\n\n<p>, which rewrites the integrand as&nbsp;<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>1u2+1<em>u<\/em>2+11\u200b<\/code><\/pre>\n\n\n\n<p>.<\/p>\n\n\n\n<p>The value of the integral is:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>\u222bex(e2x)+12ex+37dx=arctan\u2061(ex+6)+C\u222b(<em>e<\/em>2<em>x<\/em>)+12<em>e<\/em><em>x<\/em>+37<em>e<\/em><em>x<\/em>\u200b<em>d<\/em><em>x<\/em>=arctan(<em>e<\/em><em>x<\/em>+6)+<em>C<\/em><\/code><\/pre>\n\n\n\n<h3 class=\"wp-block-heading\">Explanation:<\/h3>\n\n\n\n<p>The given integral is&nbsp;<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>\u222bex(e2x)+12ex+37dx\u222b(<em>e<\/em>2<em>x<\/em>)+12<em>e<\/em><em>x<\/em>+37<em>e<\/em><em>x<\/em>\u200b<em>d<\/em><em>x<\/em><\/code><\/pre>\n\n\n\n<p>.<\/p>\n\n\n\n<p>To evaluate this integral, an appropriate substitution is the first step. Observing the numerator,&nbsp;<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>ex<em>e<\/em><em>x<\/em><\/code><\/pre>\n\n\n\n<p>, which is the derivative of&nbsp;<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>ex<em>e<\/em><em>x<\/em><\/code><\/pre>\n\n\n\n<p>, suggests a substitution involving&nbsp;<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>ex<em>e<\/em><em>x<\/em><\/code><\/pre>\n\n\n\n<p>. The denominator is a quadratic in terms of&nbsp;<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>ex<em>e<\/em><em>x<\/em><\/code><\/pre>\n\n\n\n<p>, since&nbsp;<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>e2x=(ex)2<em>e<\/em>2<em>x<\/em>=(<em>e<\/em><em>x<\/em>)2<\/code><\/pre>\n\n\n\n<p>. This structure points towards a method involving completing the square followed by an arctangent integration formula.<\/p>\n\n\n\n<p>Let&#8217;s begin by focusing on the denominator:&nbsp;<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>e2x+12ex+37<em>e<\/em>2<em>x<\/em>+12<em>e<\/em><em>x<\/em>+37<\/code><\/pre>\n\n\n\n<p>. To complete the square for the terms involving&nbsp;<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>ex<em>e<\/em><em>x<\/em><\/code><\/pre>\n\n\n\n<p>, one takes half of the coefficient of&nbsp;<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>ex<em>e<\/em><em>x<\/em><\/code><\/pre>\n\n\n\n<p>&nbsp;and squares it. The coefficient is 12, half of which is 6, and its square is 36. The denominator can be rewritten by adding and subtracting 36:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>(e2x+12ex+36)\u221236+37(<em>e<\/em>2<em>x<\/em>+12<em>e<\/em><em>x<\/em>+36)\u221236+37<\/code><\/pre>\n\n\n\n<p>The expression in the parenthesis is a perfect square:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>(ex+6)2+1(<em>e<\/em><em>x<\/em>+6)2+1<\/code><\/pre>\n\n\n\n<p>Now, the integral becomes:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>\u222bex(ex+6)2+1dx\u222b(<em>e<\/em><em>x<\/em>+6)2+1<em>e<\/em><em>x<\/em>\u200b<em>d<\/em><em>x<\/em><\/code><\/pre>\n\n\n\n<p>This form suggests the substitution&nbsp;<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>u=ex+6<em>u<\/em>=<em>e<\/em><em>x<\/em>+6<\/code><\/pre>\n\n\n\n<p>. The differential of&nbsp;<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>u<em>u<\/em><\/code><\/pre>\n\n\n\n<p>&nbsp;is found by differentiating with respect to&nbsp;<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>x<em>x<\/em><\/code><\/pre>\n\n\n\n<p>:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>dudx=ex\u27f9du=exdx<em>d<\/em><em>x<\/em><em>d<\/em><em>u<\/em>\u200b=<em>e<\/em><em>x<\/em>\u27f9<em>d<\/em><em>u<\/em>=<em>e<\/em><em>x<\/em><em>d<\/em><em>x<\/em><\/code><\/pre>\n\n\n\n<p>This substitution perfectly transforms the integral. The numerator,&nbsp;<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>exdx<em>e<\/em><em>x<\/em><em>d<\/em><em>x<\/em><\/code><\/pre>\n\n\n\n<p>, becomes&nbsp;<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>du<em>d<\/em><em>u<\/em><\/code><\/pre>\n\n\n\n<p>, and the denominator,&nbsp;<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>(ex+6)2+1(<em>e<\/em><em>x<\/em>+6)2+1<\/code><\/pre>\n\n\n\n<p>, becomes&nbsp;<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>u2+1<em>u<\/em>2+1<\/code><\/pre>\n\n\n\n<p>. The integrand is now rewritten as&nbsp;<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>1u2+1<em>u<\/em>2+11\u200b<\/code><\/pre>\n\n\n\n<p>.<\/p>\n\n\n\n<p>The integral in terms of&nbsp;<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>u<em>u<\/em><\/code><\/pre>\n\n\n\n<p>&nbsp;is:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>\u222b1u2+1du\u222b<em>u<\/em>2+11\u200b<em>d<\/em><em>u<\/em><\/code><\/pre>\n\n\n\n<p>This is a standard integral form whose antiderivative is the arctangent function. Specifically,&nbsp;<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>\u222b1u2+a2du=1aarctan\u2061(ua)+C\u222b<em>u<\/em>2+<em>a<\/em>21\u200b<em>d<\/em><em>u<\/em>=<em>a<\/em>1\u200barctan(<em>a<\/em><em>u<\/em>\u200b)+<em>C<\/em><\/code><\/pre>\n\n\n\n<p>. For this problem,&nbsp;<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>a=1<em>a<\/em>=1<\/code><\/pre>\n\n\n\n<p>.<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>\u222b1u2+1du=arctan\u2061(u)+C\u222b<em>u<\/em>2+11\u200b<em>d<\/em><em>u<\/em>=arctan(<em>u<\/em>)+<em>C<\/em><\/code><\/pre>\n\n\n\n<p>The final step is to substitute back in terms of the original variable,&nbsp;<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>x<em>x<\/em><\/code><\/pre>\n\n\n\n<p>. Replacing&nbsp;<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>u<em>u<\/em><\/code><\/pre>\n\n\n\n<p>&nbsp;with&nbsp;<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>ex+6<em>e<\/em><em>x<\/em>+6<\/code><\/pre>\n\n\n\n<p>&nbsp;gives the final result.<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>arctan\u2061(ex+6)+Carctan(<em>ex<\/em>+6)+<em>C<\/em><\/code><\/pre>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" src=\"https:\/\/learnexams.com\/blog\/wp-content\/uploads\/2025\/06\/learnexams-banner5-276.jpeg\" alt=\"\" class=\"wp-image-232072\"\/><\/figure>\n","protected":false},"excerpt":{"rendered":"<p>Consider the following integral.Find a substitution to rewrite the integrand as . Evaluate the given integral. (Use C for the constant of integration. Remember to use absolute values where appropriate.) The Correct Answer and Explanation is: Answer:The required substitution is&nbsp; , which rewrites the integrand as&nbsp; . The value of the integral is: Explanation: The [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","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":"default","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":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-232071","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/232071","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=232071"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/232071\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=232071"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=232071"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=232071"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}