{"id":8955,"date":"2024-01-30T17:51:23","date_gmt":"2024-01-30T12:21:23","guid":{"rendered":"https:\/\/physicscatalyst.com\/article\/?p=8955"},"modified":"2024-01-30T17:51:28","modified_gmt":"2024-01-30T12:21:28","slug":"integration-of-cos-cube-x","status":"publish","type":"post","link":"https:\/\/physicscatalyst.com\/article\/integration-of-cos-cube-x\/","title":{"rendered":"integration of cos cube x"},"content":{"rendered":"\n

The integration of cos cube x $\\cos ^3 x$, can be found using integration substitution and trigonometry identities . The integral of $\\cos ^3 x$ with respect to (x) is:<\/p>\n\n\n\n

\\[
\\int \\cos^3 x \\, dx =\\frac {\\sin 3x}{12} + \\frac {3\\sin x}{4} + C
\\]<\/p>\n\n\n\n

or<\/p>\n\n\n\n

\\[
\\int \\cos^3 x \\, dx =-\\frac{\\sin^3(x)}{3} + \\sin(x) + C
\\]<\/p>\n\n\n\n

Here, (C) represents the constant of integration, which is added because the process of integration determines the antiderivative up to an arbitrary constant.<\/p>\n\n\n\n

Proof of integration of cos cube x<\/h2>\n\n\n\n

Method of trigonometry identities<\/strong><\/p>\n\n\n\n

(I) <\/p>\n\n\n\n

we know that <\/p>\n\n\n\n

$\\cos(3x)=4\\cos^{3}x-3\\cos(x)$
$\\cos^{3}x= \\frac {\\cos 3x + 3 \\cos x}{4}$<\/p>\n\n\n\n

Therefore<\/p>\n\n\n\n

\\[
\\int \\cos^3 x \\, dx =\\int \\frac {\\cos 3x + 3 \\cos x}{4} \\, dx
\\]<\/p>\n\n\n\n

Now we know that by integration by substitution<\/p>\n\n\n\n

$\\\\int cos nx = \\frac {\\sin nx}{n}<\/p>\n\n\n\n

Therefore<\/p>\n\n\n\n

\\[
\\int \\cos^3 x \\, dx=\\frac {\\sin 3x}{12} + \\frac {3\\sin x}{4} + C
\\]<\/p>\n\n\n\n

(II)<\/p>\n\n\n\n

Now we know that<\/p>\n\n\n\n

$\\sin(3x)=3\\sin(x)-4\\sin^{3}x$ <\/p>\n\n\n\n

Substituting this back in above we get<\/p>\n\n\n\n

\\[
\\int \\cos^3 x \\, dx =-\\frac{\\sin^3(x)}{3} + \\sin(x) + C
\\]<\/p>\n\n\n\n

Method of integration by substitution<\/strong><\/p>\n\n\n\n

\\[
\\int \\cos^3 x \\, dx =\\int cos^2 x cos x \\, dx = \\int (1-sin^2 x) cos x \\, dx
\\]<\/p>\n\n\n\n

Now let $u= sin x$ ,then $ du= cos x dx$
Therefore<\/p>\n\n\n\n

\\[
= \\int (1-u^2) \\, du = u – \\frac {u^3}{3}
\\]<\/p>\n\n\n\n

Substituting back<\/p>\n\n\n\n

\\[
\\int \\cos^3 x \\, dx =-\\frac{\\sin^3(x)}{3} + \\sin(x) + C
\\]<\/p>\n\n\n\n

we can use the identity to convert back into form (I)<\/p>\n\n\n\n

$\\sin(3x)=3\\sin(x)-4\\sin^{3}x$ <\/p>\n\n\n\n

\n

Other Integration Related Articles<\/p>\n\n\n\n

Integration of modulus function<\/a><\/td>Integration of tan x<\/a><\/td><\/td><\/td><\/tr>
Integration of sec x<\/a><\/td>Integration of greatest integer function<\/a><\/td><\/td><\/td><\/tr>
Integration of trigonometric functions<\/a><\/td>Integration of cot x<\/a><\/td><\/td><\/td><\/tr>
Integration of cosec x<\/a><\/td>Integration of sinx<\/a><\/td><\/td><\/td><\/tr>
Integration of irrational functions<\/a><\/td>integration of root x<\/a><\/td><\/td><\/td><\/tr>
Integration of fractional part of x<\/a><\/td>Integration of cos square x<\/a><\/td><\/td><\/td><\/tr><\/tbody><\/table><\/figure>\n<\/div><\/div>\n","protected":false},"excerpt":{"rendered":"

The integration of cos cube x $\\cos ^3 x$, can be found using integration substitution and trigonometry identities . The integral of $\\cos ^3 x$ with respect to (x) is: \\[\\int \\cos^3 x \\, dx =\\frac {\\sin 3x}{12} + \\frac {3\\sin x}{4} + C\\] or \\[\\int \\cos^3 x \\, dx =-\\frac{\\sin^3(x)}{3} + \\sin(x) + C\\] […]<\/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-8955","post","type-post","status-publish","format-standard","hentry","category-maths"],"yoast_head":"\nintegration of cos cube x - physicscatalyst's Blog<\/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:\/\/physicscatalyst.com\/article\/integration-of-cos-cube-x\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"integration of cos cube x - physicscatalyst's Blog\" \/>\n<meta property=\"og:description\" content=\"The integration of cos cube x $cos ^3 x$, can be found using integration substitution and trigonometry identities . The integral of $cos ^3 x$ with respect to (x) is: [int cos^3 x , dx =frac {sin 3x}{12} + frac {3sin x}{4} + C] or [int cos^3 x , dx =-frac{sin^3(x)}{3} + sin(x) + C] […]\" \/>\n<meta property=\"og:url\" content=\"https:\/\/physicscatalyst.com\/article\/integration-of-cos-cube-x\/\" \/>\n<meta property=\"og:site_name\" content=\"physicscatalyst'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=\"2024-01-30T12:21:23+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2024-01-30T12:21:28+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=\"2 minutes\" \/>\n<!-- \/ Yoast SEO plugin. -->","yoast_head_json":{"title":"integration of cos cube x - physicscatalyst's Blog","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-cos-cube-x\/","og_locale":"en_US","og_type":"article","og_title":"integration of cos cube x - physicscatalyst's Blog","og_description":"The integration of cos cube x $cos ^3 x$, can be found using integration substitution and trigonometry identities . 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The integral of $\\cos ^3 x$ with respect to (x) is: \\[\\int \\cos^3 x \\, dx =\\frac {\\sin 3x}{12} + \\frac {3\\sin x}{4} + C\\] or \\[\\int \\cos^3 x \\, dx =-\\frac{\\sin^3(x)}{3} + \\sin(x) + C\\]…","_links":{"self":[{"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/posts\/8955","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=8955"}],"version-history":[{"count":2,"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/posts\/8955\/revisions"}],"predecessor-version":[{"id":8979,"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/posts\/8955\/revisions\/8979"}],"wp:attachment":[{"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/media?parent=8955"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/categories?post=8955"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/tags?post=8955"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}