Integration of cos square x can be calculated using trigonometric identities .Here is the formula for it<\/p>\n\n\n\n
$$ \\int \\cos^2(x) \\, dx = \\frac{1}{2} x + \\frac{1}{4} \\sin(2x) + C $$<\/p>\n\n\n\n
To integrate $ \\cos^2(x) $, we use a trigonometric identity to simplify the expression before integrating. The identity commonly used is the power-reduction formula or double angle identity<\/p>\n\n\n\n
$$ \\cos^2(x) = \\frac{1 + \\cos(2x)}{2} $$<\/p>\n\n\n\n
Now, let’s integrate using this identity:<\/p>\n\n\n\n
$$ \\int \\cos^2(x) \\, dx = \\int \\frac{1 + \\cos(2x)}{2} \\, dx $$<\/p>\n\n\n\n
This integral can be split into two simpler integrals:<\/p>\n\n\n\n
$$ = \\frac{1}{2} \\int dx + \\frac{1}{2} \\int \\cos(2x) \\, dx $$<\/p>\n\n\n\n
Now, integrate each part:<\/p>\n\n\n\n
Proof of this Integral<\/strong> $\\int \\cos(2x) dx= \\frac{\\sin(2x)}{2}$<\/p>\n\n\n\n Let t=2x So, the integral becomes:<\/p>\n\n\n\n $$ = \\frac{1}{2} x + \\frac{1}{4} \\sin(2x) + C $$<\/p>\n\n\n\n where $ C $ is the constant of integration. Therefore, the integral of $ \\cos^2(x) $ is:<\/p>\n\n\n\n $$ \\int \\cos^2(x) \\, dx = \\frac{1}{2} x + \\frac{1}{4} \\sin(2x) + C $$<\/p>\n\n\n\n To find the definite integral of $ \\cos^2(x) $ over a specific interval, we use the same approach as with the indefinite integral, but we’ll apply the limits of integration at the end.<\/p>\n\n\n\n So, the definite integral of $ \\cos^2(x) $ from ( a ) to ( b ) is:<\/p>\n\n\n\n \\[ \\int_a^b \\cos^2(x) \\, dx = \\frac{1}{2} (b – a) + \\frac{1}{4} (\\sin(2b) – \\sin(2a)) \\]<\/p>\n\n\n\n Example<\/strong><\/p>\n\n\n\n $$ \\int_0^\\pi \\cos^2(x) \\, dx $$<\/p>\n\n\n\n First, we use the power-reduction formula:<\/p>\n\n\n\n $$ \\cos^2(x) = \\frac{1 + \\cos(2x)}{2} $$<\/p>\n\n\n\n Now, integrate over the interval $ [0, \\pi] $:<\/p>\n\n\n\n $$ \\int_0^\\pi \\frac{1 + \\cos(2x)}{2} \\, dx $$<\/p>\n\n\n\n $$ = \\frac{1}{2} \\int_0^\\pi dx + \\frac{1}{2} \\int_0^\\pi \\cos(2x) \\, dx $$<\/p>\n\n\n\n $$= \\frac{1}{2} x \\Big|_0^\\pi + \\frac{1}{4} \\sin(2x) \\Big|_0^\\pi $$ Therefore, the definite integral of $ \\cos^2(x) $ from $ 0 $ to $ \\pi $ is $ \\frac{\\pi}{2} $. This result represents the area under the curve of $ \\cos^2(x) $ between $ x = 0 $ and $ x = \\pi $.<\/p>\n\n\n\n Other Integration Related Articles<\/p>\n\n\n\n Integration of cos square x can be calculated using trigonometric identities .Here is the formula for it $$ \\int \\cos^2(x) \\, dx = \\frac{1}{2} x + \\frac{1}{4} \\sin(2x) + C $$ Proof of Integration of cos square x To integrate $ \\cos^2(x) $, we use a trigonometric identity to simplify the expression before integrating. The […]<\/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":[506],"class_list":["post-8684","post","type-post","status-publish","format-standard","hentry","category-general","category-maths","tag-integration"],"yoast_head":"\n
dt=2 dx
or
dx= dt\/2
Therefore
$\\int \\cos(2x) \\; dx== \\frac {1}{2} \\int cos t \\; dt= \\frac{\\sin(2x)}{2} $<\/p>\n\n\n\nDefinite Integral of cos square x<\/h2>\n\n\n\n
$$=\\frac{\\pi}{2} + 0=\\frac{\\pi}{2}$ $<\/p>\n\n\n\nIntegration 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":"