{"id":8978,"date":"2024-02-02T18:53:43","date_gmt":"2024-02-02T13:23:43","guid":{"rendered":"https:\/\/physicscatalyst.com\/article\/?p=8978"},"modified":"2024-02-02T18:53:50","modified_gmt":"2024-02-02T13:23:50","slug":"integration-of-sin-cube-x","status":"publish","type":"post","link":"https:\/\/physicscatalyst.com\/article\/integration-of-sin-cube-x\/","title":{"rendered":"integration of sin cube x"},"content":{"rendered":"\n

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

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

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

\\[
\\int \\sin^3 x \\, dx =+\\frac{\\cos^3(x)}{3} – \\cos(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 sin 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

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

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

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

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

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

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

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

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

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

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

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

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

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

We can use the identity for $\\sin^2(x) = 1 – \\cos^2(x)$ to rewrite $\\sin^3(x)$ as:
$$ \\sin^3(x) = \\sin(x) \\sin^2(x) = \\sin(x)(1 – \\cos^2(x)) $$
$$ \\sin^3(x) = \\sin(x) – \\sin(x) \\cos^2(x) $$<\/p>\n\n\n\n

Therefore
$$ \\int \\sin^3(x) \\, dx = \\int \\sin(x) \\, dx – \\int \\sin(x) \\cos^2(x) \\, dx $$<\/p>\n\n\n\n

The integral of $\\sin(x)$ is straightforward:
$$ \\int \\sin(x) \\, dx = -\\cos(x) $$<\/p>\n\n\n\n

For the integral $ \\int \\sin(x) \\cos^2(x) \\, dx $, let $ u = \\cos(x) $, which implies $ du = -\\sin(x) dx $.<\/p>\n\n\n\n

Rewriting the integral in terms of $u$:
$$ -\\int u^2 \\, du $$<\/p>\n\n\n\n

Integrate $ u^2 $:
$$ -\\int u^2 \\, du = -\\frac{u^3}{3} = -\\frac{\\cos^3(x)}{3} $$<\/p>\n\n\n\n

Combine the results:
$$ \\int \\sin^3(x) \\, dx = -\\cos(x) + \\frac{\\cos^3(x)}{3} + C $$<\/p>\n\n\n\n

So, the integral of $\\sin^3(x)$ is:
$$ -\\cos(x) + \\frac{\\cos^3(x)}{3} + C $$<\/p>\n\n\n\n

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

$\\cos(3x)=4\\cos^{3}x-3\\cos(x)$<\/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 sin cube x $\\sin ^3 x$, can be found using integration substitution and trigonometry identities . The integral of $\\sin ^3 x$ with respect to (x) is: \\[\\int \\sin^3 x \\, dx =\\frac {\\cos 3x}{12} – \\frac {3\\cos x}{4} + C\\] or \\[\\int \\sin^3 x \\, dx =+\\frac{\\cos^3(x)}{3} – \\cos(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":[24,498],"tags":[],"class_list":["post-8978","post","type-post","status-publish","format-standard","hentry","category-general","category-maths"],"yoast_head":"\nintegration of sin 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-sin-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 sin cube x - physicscatalyst's Blog\" \/>\n<meta property=\"og:description\" content=\"The integration of sin cube x $sin ^3 x$, can be found using integration substitution and trigonometry identities . The integral of $sin ^3 x$ with respect to (x) is: [int sin^3 x , dx =frac {cos 3x}{12} – frac {3cos x}{4} + C] or [int sin^3 x , dx =+frac{cos^3(x)}{3} – cos(x) + C] […]\" \/>\n<meta property=\"og:url\" content=\"https:\/\/physicscatalyst.com\/article\/integration-of-sin-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=\"2024-02-02T13:23:43+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2024-02-02T13:23:50+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 sin 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-sin-cube-x\/","og_locale":"en_US","og_type":"article","og_title":"integration of sin cube x - physicscatalyst's Blog","og_description":"The integration of sin cube x $sin ^3 x$, can be found using integration substitution and trigonometry identities . 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The integral of $\\sin ^3 x$ with respect to (x) is: \\[\\int \\sin^3 x \\, dx =\\frac {\\cos 3x}{12} – \\frac {3\\cos x}{4} + C\\] or \\[\\int \\sin^3 x \\, dx =+\\frac{\\cos^3(x)}{3} – \\cos(x) + C\\]…","_links":{"self":[{"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/posts\/8978","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=8978"}],"version-history":[{"count":2,"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/posts\/8978\/revisions"}],"predecessor-version":[{"id":8981,"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/posts\/8978\/revisions\/8981"}],"wp:attachment":[{"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/media?parent=8978"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/categories?post=8978"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/tags?post=8978"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}