{"id":8843,"date":"2024-01-19T19:08:44","date_gmt":"2024-01-19T13:38:44","guid":{"rendered":"https:\/\/physicscatalyst.com\/article\/?p=8843"},"modified":"2024-01-23T15:36:29","modified_gmt":"2024-01-23T10:06:29","slug":"integration-of-even-function","status":"publish","type":"post","link":"https:\/\/physicscatalyst.com\/article\/integration-of-even-function\/","title":{"rendered":"Integration of even function"},"content":{"rendered":"\n<ul class=\"wp-block-list\">\n<li>An even function $ f(x) $ satisfies the condition $ f(x) = f(-x) $ for all $ x $ in its domain .This symmetry means that the function&#8217;s graph is the same on both sides of the y-axis.<\/li>\n\n\n\n<li>Therefore The integration of  even function over a symmetric interval can be simplified due to the symmetry of the function. .<\/li>\n<\/ul>\n\n\n\n<p>Integration of even function over a symmetric intervalcan be simplified as follows:<\/p>\n\n\n\n<p>$<br>\\int_{-a}^{a} f(x) \\, dx = 2 \\int_{0}^{a} f(x) \\, dx<br>$<\/p>\n\n\n\n<p>This simplification is possible because the area under the curve from $-a$ to $0$ is the same as the area from $0$ to $a$ due to the symmetry of the function.<\/p>\n\n\n\n<p><strong>Example 1<\/strong><br>$<br>\\int_{-a}^{a} x^2 \\, dx<br>$<\/p>\n\n\n\n<p><strong>Solution<\/strong><br>Since $ x^2 $ is an even function, the integral can be simplified to:<\/p>\n\n\n\n<p>$<br>2 \\int_{0}^{a} x^2 \\, dx<br>$<\/p>\n\n\n\n<p>Calculating this integral gives:<\/p>\n\n\n\n<p>$<br>= 2 \\left[ \\frac{x^3}{3} \\right]_{0}^{a}<br>= 2 \\left( \\frac{a^3}{3} &#8211; 0 \\right)<br>= \\frac{2a^3}{3}<br>$<\/p>\n\n\n\n<p><strong>Example 2<\/strong><br>$<br>\\int_{-\\pi\/2}^{\\pi\/2} \\cos^2(x) \\, dx<br>$<\/p>\n\n\n\n<p><strong>Solution<\/strong><br>The function $ \\cos^2(x) $ is an even function because $ \\cos^2(x) = \\cos^2(-x) $.<\/p>\n\n\n\n<p>\\ Since $ f(x) $ is even, we can simplify the integral:<\/p>\n\n\n\n<p>$<br>\\int_{-\\pi\/2}^{\\pi\/2} \\cos^2(x) \\, dx = 2 \\int_{0}^{\\pi\/2} \\cos^2(x) \\, dx<br>$<\/p>\n\n\n\n<p>To integrate $ \\cos^2(x) $, use the identity:<\/p>\n\n\n\n<p>$<br>\\cos^2(x) = \\frac{1}{2}(1 + \\cos(2x))<br>$<\/p>\n\n\n\n<p>Thus, the integral becomes:<\/p>\n\n\n\n<p>$<br>2 \\int_{0}^{\\pi\/2} \\frac{1}{2}(1 + \\cos(2x)) \\, dx<br>$<\/p>\n\n\n\n<p>$<br>= \\int_{0}^{\\pi\/2} (1 + \\cos(2x)) \\, dx<br>= \\left[ x + \\frac{1}{2} \\sin(2x) \\right]_{0}^{\\pi\/2}<br>$<\/p>\n\n\n\n<p>$<br>= \\left( \\frac{\\pi}{2} + 0 \\right) &#8211; \\left( 0 + 0 \\right)<br>= \\frac{\\pi}{2}<br>$<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Solved examples on Integration of even function<\/h2>\n\n\n\n<p><strong>Question  1<\/strong><br>$<br>\\int_{-1}^{1} x^4 &#8211; 6x^2 + 8 \\, dx<br>$<\/p>\n\n\n\n<p><strong>Solution<\/strong><br>The function $ f(x) = x^4 &#8211; 6x^2 + 8 $ is even since all terms are even powers of $ x $.<\/p>\n\n\n\n<p>$<br>\\int_{-1}^{1} (x^4 &#8211; 6x^2 + 8) \\, dx = 2 \\int_{0}^{1} (x^4 &#8211; 6x^2 + 8) \\, dx<br>$<\/p>\n\n\n\n<p>$<br>= 2 \\left[ \\frac{x^5}{5} &#8211; 2x^3 + 8x \\right]_{0}^{1}<br>$<\/p>\n\n\n\n<p>$<br>= 2 \\left( \\frac{1^5}{5} &#8211; 2 \\cdot 1^3 + 8 \\cdot 1 \\right) &#8211; 0 =62\/5<br>$<\/p>\n\n\n\n<p><strong>Question 2<\/strong><\/p>\n\n\n\n<p>$<br>\\int_{-\\pi}^{\\pi} \\sin^2(x) \\, dx<br>$<br><strong>Solution<\/strong><\/p>\n\n\n\n<p>$ \\sin^2(x) $ is even because $ \\sin^2(x) = \\sin^2(-x) $.<\/p>\n\n\n\n<p>$<br>\\int_{-\\pi}^{\\pi} \\sin^2(x) \\, dx = 2 \\int_{0}^{\\pi} \\sin^2(x) \\, dx<br>$<\/p>\n\n\n\n<p>$<br>\\sin^2(x) = \\frac{1}{2}(1 &#8211; \\cos(2x))<br>$<\/p>\n\n\n\n<p>$<br>= 2 \\int_{0}^{\\pi} \\frac{1}{2}(1 &#8211; \\cos(2x)) \\, dx<br>= \\int_{0}^{\\pi} (1 &#8211; \\cos(2x)) \\, dx<br>$<\/p>\n\n\n\n<p>$<br>= \\left[ x &#8211; \\frac{1}{2} \\sin(2x) \\right]_{0}^{\\pi} = \\pi<br>$<\/p>\n\n\n\n<p><strong>Question 3<\/strong><\/p>\n\n\n\n<p>$<br>\\int_{-1}^{1} \\frac {x^3 + |x| + 1}{x^2 + 2|x| +1} \\, dx<br>$<\/p>\n\n\n\n<p><strong>Solution<\/strong><br>$<br>\\int_{-1}^{1} \\frac {x^3 + |x| + 1}{x^2 + 2|x| +1} \\, dx = \\int_{-1}^{1} \\frac {x^3}{x^2 + 2|x| +1} \\, dx + \\int_{-1}^{1} \\frac {|x| + 1}{x^2 + 2|x| +1} \\, dx<br>$<\/p>\n\n\n\n<p>Now<br>$\\int_{-1}^{1} \\frac {x^3}{x^2 + 2|x| +1} \\, dx = 0$ as this is odd function<br>$\\int_{-1}^{1} \\frac {|x| + 1}{x^2 + 2|x| +1} \\, dx= 2\\int_{-1}^{1} \\frac {|x| + 1}{x^2 + 2|x| +1} \\, dx$ as this is even function<br>Hence<\/p>\n\n\n\n<p>$<br>=\\int_{-1}^{1} \\frac { |x| + 1}{x^2 + 2|x| +1} \\, dx \\\\<br>= 2 \\int_{0}^{1} \\frac { |x| + 1}{x^2 + 2|x| +1} \\, dx \\\\<br>=2 \\int_{0}^{1} \\frac {|x|+1}{(|x|+1)^2} \\, dx \\\\<br>=2 \\int_{0}^{1} \\frac {1}{|x| + 1} \\, dx \\\\<br>= 2 \\int_{0}^{1} \\frac {1}{x + 1} \\, dx \\\\<br>=2 \\log 2<br>$<\/p>\n\n\n\n<p>I hope you find this article on Integration of even function interesting . Please do provide the feedback<\/p>\n\n\n\n<div class=\"wp-block-group has-ast-global-color-4-background-color has-background is-layout-constrained wp-container-core-group-is-layout-ca99af60 wp-block-group-is-layout-constrained\" style=\"border-radius:17px;margin-top:var(--wp--preset--spacing--20);margin-bottom:var(--wp--preset--spacing--20);padding-top:var(--wp--preset--spacing--20);padding-right:var(--wp--preset--spacing--50);padding-bottom:var(--wp--preset--spacing--20);padding-left:var(--wp--preset--spacing--50)\"><div class=\"wp-block-group__inner-container\">\n<p class=\"has-medium-font-size\">Other Integration Related Articles<\/p>\n\n\n\n<figure class=\"wp-block-table is-style-regular\" style=\"padding-right:0;padding-left:0;font-size:16px\"><table style=\"border-style:none;border-width:0px\"><tbody><tr><td><a href=\"https:\/\/physicscatalyst.com\/article\/integration-of-tan-inverse-x\/\">Integration of tan inverse x<\/a><\/td><td><a href=\"https:\/\/physicscatalyst.com\/article\/integration-of-cos-inverse-x\/\">Integration of cos inverse x<\/a><\/td><\/tr><tr><td><a href=\"https:\/\/physicscatalyst.com\/article\/integration-of-cos-square-x\/\">Integration of cos square x<\/a><\/td><td><a href=\"https:\/\/physicscatalyst.com\/article\/integration-of-fractional-part-of-x\/\">Integration of fractional part of x<\/a><\/td><\/tr><tr><td><a href=\"https:\/\/physicscatalyst.com\/article\/integration-of-uv-formula\/\">integration of uv formula<\/a><\/td><td><a href=\"https:\/\/physicscatalyst.com\/article\/integration-of-log-x\/\">integration of log x<\/a><\/td><\/tr><tr><td><a href=\"https:\/\/physicscatalyst.com\/article\/integration-of-sin-inverse-x\/\">integration of sin inverse x<\/a><\/td><td><a href=\"https:\/\/physicscatalyst.com\/article\/integration-of-constant\/\">Integration of constant term<\/a><\/td><\/tr><tr><td><a href=\"https:\/\/physicscatalyst.com\/article\/integration-of-hyperbolic-functions\/\">integration of hyperbolic functions<\/a><\/td><td><a data-type=\"post\" data-id=\"5701\" href=\"https:\/\/physicscatalyst.com\/article\/integration-formulas-list\/\">List of Integration Formulas <\/a><\/td><\/tr><tr><td><a href=\"https:\/\/physicscatalyst.com\/article\/integration-of-log-sinx\/\">integration of log sinx<\/a><\/td><td><a href=\"https:\/\/physicscatalyst.com\/article\/integration-of-sin-x-cos-x-dx\/\">integration of sin x cos x dx<\/a><\/td><\/tr><\/tbody><\/table><\/figure>\n<\/div><\/div>\n","protected":false},"excerpt":{"rendered":"<p>Integration of even function over a symmetric intervalcan be simplified as follows: $\\int_{-a}^{a} f(x) \\, dx = 2 \\int_{0}^{a} f(x) \\, dx$ This simplification is possible because the area under the curve from $-a$ to $0$ is the same as the area from $0$ to $a$ due to the symmetry of the function. Example 1$\\int_{-a}^{a} [&hellip;]<\/p>\n","protected":false},"author":8,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_uag_custom_page_level_css":"","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":"","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":""}},"footnotes":""},"categories":[498],"tags":[],"class_list":["post-8843","post","type-post","status-publish","format-standard","hentry","category-maths"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.4 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Integration of even function - physicscatalyst&#039;s Blog<\/title>\n<meta name=\"description\" content=\"Integration of even function over symmetric limits simplifies as areas on both sides of the y-axis mirror, easing calculations\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/physicscatalyst.com\/article\/integration-of-even-function\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Integration of even function - physicscatalyst&#039;s Blog\" \/>\n<meta property=\"og:description\" content=\"Integration of even function over symmetric limits simplifies as areas on both sides of the y-axis mirror, easing calculations\" \/>\n<meta property=\"og:url\" content=\"https:\/\/physicscatalyst.com\/article\/integration-of-even-function\/\" \/>\n<meta property=\"og:site_name\" content=\"physicscatalyst&#039;s Blog\" \/>\n<meta property=\"article:publisher\" content=\"https:\/\/www.facebook.com\/PhysicsCatalyst\" \/>\n<meta property=\"article:author\" content=\"https:\/\/www.facebook.com\/PhysicsCatalyst\" \/>\n<meta property=\"article:published_time\" content=\"2024-01-19T13:38:44+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2024-01-23T10:06:29+00:00\" \/>\n<meta name=\"author\" content=\"physicscatalyst\" \/>\n<meta name=\"twitter:label1\" content=\"Written by\" \/>\n\t<meta name=\"twitter:data1\" content=\"physicscatalyst\" \/>\n\t<meta name=\"twitter:label2\" content=\"Est. reading time\" \/>\n\t<meta name=\"twitter:data2\" content=\"3 minutes\" \/>\n<!-- \/ Yoast SEO plugin. -->","yoast_head_json":{"title":"Integration of even function - physicscatalyst&#039;s Blog","description":"Integration of even function over symmetric limits simplifies as areas on both sides of the y-axis mirror, easing calculations","robots":{"index":"index","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"canonical":"https:\/\/physicscatalyst.com\/article\/integration-of-even-function\/","og_locale":"en_US","og_type":"article","og_title":"Integration of even function - physicscatalyst&#039;s Blog","og_description":"Integration of even function over symmetric limits simplifies as areas on both sides of the y-axis mirror, easing calculations","og_url":"https:\/\/physicscatalyst.com\/article\/integration-of-even-function\/","og_site_name":"physicscatalyst&#039;s Blog","article_publisher":"https:\/\/www.facebook.com\/PhysicsCatalyst","article_author":"https:\/\/www.facebook.com\/PhysicsCatalyst","article_published_time":"2024-01-19T13:38:44+00:00","article_modified_time":"2024-01-23T10:06:29+00:00","author":"physicscatalyst","twitter_misc":{"Written by":"physicscatalyst","Est. reading time":"3 minutes"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"Article","@id":"https:\/\/physicscatalyst.com\/article\/integration-of-even-function\/#article","isPartOf":{"@id":"https:\/\/physicscatalyst.com\/article\/integration-of-even-function\/"},"author":{"name":"physicscatalyst","@id":"https:\/\/physicscatalyst.com\/article\/#\/schema\/person\/9b302efdc9b32e459cb1e61ab7506d3f"},"headline":"Integration of even function","datePublished":"2024-01-19T13:38:44+00:00","dateModified":"2024-01-23T10:06:29+00:00","mainEntityOfPage":{"@id":"https:\/\/physicscatalyst.com\/article\/integration-of-even-function\/"},"wordCount":459,"commentCount":0,"publisher":{"@id":"https:\/\/physicscatalyst.com\/article\/#organization"},"articleSection":["Maths"],"inLanguage":"en-US","potentialAction":[{"@type":"CommentAction","name":"Comment","target":["https:\/\/physicscatalyst.com\/article\/integration-of-even-function\/#respond"]}]},{"@type":"WebPage","@id":"https:\/\/physicscatalyst.com\/article\/integration-of-even-function\/","url":"https:\/\/physicscatalyst.com\/article\/integration-of-even-function\/","name":"Integration of even function - physicscatalyst&#039;s Blog","isPartOf":{"@id":"https:\/\/physicscatalyst.com\/article\/#website"},"datePublished":"2024-01-19T13:38:44+00:00","dateModified":"2024-01-23T10:06:29+00:00","description":"Integration of even function over symmetric limits simplifies as areas on both sides of the y-axis mirror, easing calculations","breadcrumb":{"@id":"https:\/\/physicscatalyst.com\/article\/integration-of-even-function\/#breadcrumb"},"inLanguage":"en-US","potentialAction":[{"@type":"ReadAction","target":["https:\/\/physicscatalyst.com\/article\/integration-of-even-function\/"]}]},{"@type":"BreadcrumbList","@id":"https:\/\/physicscatalyst.com\/article\/integration-of-even-function\/#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https:\/\/physicscatalyst.com\/article\/"},{"@type":"ListItem","position":2,"name":"Maths","item":"https:\/\/physicscatalyst.com\/article\/maths\/"},{"@type":"ListItem","position":3,"name":"Integration of even function"}]},{"@type":"WebSite","@id":"https:\/\/physicscatalyst.com\/article\/#website","url":"https:\/\/physicscatalyst.com\/article\/","name":"physicscatalyst's Blog","description":"Learn free for class 9th, 10th science\/maths , 12th and IIT-JEE Physics and maths.","publisher":{"@id":"https:\/\/physicscatalyst.com\/article\/#organization"},"potentialAction":[{"@type":"SearchAction","target":{"@type":"EntryPoint","urlTemplate":"https:\/\/physicscatalyst.com\/article\/?s={search_term_string}"},"query-input":{"@type":"PropertyValueSpecification","valueRequired":true,"valueName":"search_term_string"}}],"inLanguage":"en-US"},{"@type":"Organization","@id":"https:\/\/physicscatalyst.com\/article\/#organization","name":"physicscatalyst","url":"https:\/\/physicscatalyst.com\/article\/","logo":{"@type":"ImageObject","inLanguage":"en-US","@id":"https:\/\/physicscatalyst.com\/article\/#\/schema\/logo\/image\/","url":"https:\/\/physicscatalyst.com\/article\/wp-content\/uploads\/2024\/08\/cropped-logo-1.jpg","contentUrl":"https:\/\/physicscatalyst.com\/article\/wp-content\/uploads\/2024\/08\/cropped-logo-1.jpg","width":96,"height":96,"caption":"physicscatalyst"},"image":{"@id":"https:\/\/physicscatalyst.com\/article\/#\/schema\/logo\/image\/"},"sameAs":["https:\/\/www.facebook.com\/PhysicsCatalyst","https:\/\/x.com\/physicscatalyst","https:\/\/www.youtube.com\/user\/thephysicscatalyst","https:\/\/www.instagram.com\/physicscatalyst\/"]},{"@type":"Person","@id":"https:\/\/physicscatalyst.com\/article\/#\/schema\/person\/9b302efdc9b32e459cb1e61ab7506d3f","name":"physicscatalyst","sameAs":["https:\/\/physicscatalyst.com","https:\/\/www.facebook.com\/PhysicsCatalyst","https:\/\/x.com\/physicscatalyst"]}]}},"uagb_featured_image_src":{"full":false,"thumbnail":false,"medium":false,"medium_large":false,"large":false,"1536x1536":false,"2048x2048":false,"shareaholic-thumbnail":false},"uagb_author_info":{"display_name":"physicscatalyst","author_link":"https:\/\/physicscatalyst.com\/article\/author\/physicscatalyst\/"},"uagb_comment_info":0,"uagb_excerpt":"Integration of even function over a symmetric intervalcan be simplified as follows: $\\int_{-a}^{a} f(x) \\, dx = 2 \\int_{0}^{a} f(x) \\, dx$ This simplification is possible because the area under the curve from $-a$ to $0$ is the same as the area from $0$ to $a$ due to the symmetry of the function. Example 1$\\int_{-a}^{a}&hellip;","_links":{"self":[{"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/posts\/8843","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/users\/8"}],"replies":[{"embeddable":true,"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/comments?post=8843"}],"version-history":[{"count":3,"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/posts\/8843\/revisions"}],"predecessor-version":[{"id":8857,"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/posts\/8843\/revisions\/8857"}],"wp:attachment":[{"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/media?parent=8843"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/categories?post=8843"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/tags?post=8843"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}