Integration of sin x cos x dx can be found using various integration technique like integration by substitution, Integration by partial fraction along with trigonometric identities. The various formula for integration of sin x cos x dx are<\/p>\n\n\n\n
I<\/p>\n\n\n\n
\\[
\\int \\sin(x) \\cos (x)\\, dx = -\\frac {1}{4} \\cos (2x) + C
\\]<\/p>\n\n\n\n
II<\/p>\n\n\n\n
\\[
\\int \\sin(x) \\cos (x)\\, dx = -\\frac {1}{2} \\cos^2 (x) + C
\\]<\/p>\n\n\n\n
III<\/p>\n\n\n\n
\\[
\\int \\sin(x) \\cos (x)\\, dx = \\frac {1}{2} \\sin^2 (x) + C
\\]<\/p>\n\n\n\n
Lets check out the proof of each of these integration of sin x cos x dx<\/p>\n\n\n\n
Integration of cos x sin x can be solved by using a clever trick involving multiplying and dividing by 2. Here’s how it’s done:<\/p>\n\n\n\n
$ \\int \\sin(x) \\cos (x)\\, dx = \\frac {1}{2} \\int 2\\sin(x) \\cos (x)\\, dx = \\frac {1}{2} \\int \\sin (2x) \\, dx$<\/p>\n\n\n\n
Now taking 2x= t
$2 dx= dt$
$dx= \\frac {dt}{2}$<\/p>\n\n\n\n
$ \\int \\sin(x) \\cos (x)\\, dx = \\frac {1}{4} \\int \\sin (t) \\; dt = -\\frac {1}{4} \\cos (2x) + C$<\/p>\n\n\n\n
where (C) is the constant of integration. This is the integral of $(\\sin(x) \\cos(x))$.<\/p>\n\n\n\n
$ \\int \\sin(x) \\cos (x)\\, dx$<\/p>\n\n\n\n
let $\\cos x = t$
$-\\sin x dx = dt$
substituting these<\/p>\n\n\n\n
$ \\int \\sin(x) \\cos (x)\\, dx = – \\int t \\; dt = -\\frac {t^2}{2} + C$<\/p>\n\n\n\n
Hence<\/p>\n\n\n\n
\\[
\\int \\sin(x) \\cos (x)\\, dx = -\\frac {1}{2} \\cos^2 (x) + C
\\]<\/p>\n\n\n\n
$ \\int \\sin(x) \\cos (x)\\, dx$<\/p>\n\n\n\n
let $\\sin x = t$
$\\cos x dx = dt$
substituting these<\/p>\n\n\n\n
$ \\int \\sin(x) \\cos (x)\\, dx = \\int t \\; dt = \\frac {t^2}{2} + C$<\/p>\n\n\n\n
Hence<\/p>\n\n\n\n
\\[
\\int \\sin(x) \\cos (x)\\, dx = \\frac {1}{2} \\sin^2 (x) + C
\\]
<\/p>\n\n\n\n
To find the definite integral of $\\sin x \\cos 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
The definite integral of $\\sin x \\cos x $ from $a$ to $b$ is given by:<\/p>\n\n\n\n
$$\\int_{a}^{b} \\sin(x) \\cos (x) \\, dx = – \\frac {1}{4}\\cos (2b) + \\frac {1}{4} \\cos (2a) $$<\/p>\n\n\n\n
This expression represents the accumulated area under the curve of $\\sin x \\cos x $ from $x = a$ to $x = b$<\/p>\n\n\n\n
Question 1<\/strong><\/p>\n\n\n\n $\\int_{0}^{\\pi} \\sin(x) \\cos (x) \\, dx $<\/p>\n\n\n\n Solution<\/strong><\/p>\n\n\n\n $\\int_{0}^{\\pi} \\sin(x) \\cos (x) \\, dx =- \\frac {1}{4}\\cos (2\\pi) + \\frac {1}{4} \\cos (0)= 0$<\/p>\n\n\n\n Question 2<\/strong><\/p>\n\n\n\n Evaluate the definite integral of $\\cos x \\sin x$ from $0$ to $\\pi\/4$ and then use the result to find the area enclosed between the curve $y = \\cos x \\sin x$, the x-axis, and the lines $x = 0$ and $x = \\pi\/4$.<\/p>\n\n\n\n Solution<\/strong> We can calculate this integral:<\/p>\n\n\n\n $$ $$ $$ I hope you find this article on integration of sin x cos x dx interesting and useful . Please do provide the feedback<\/p>\n\n\n\n Other Integration Related Articles<\/p>\n\n\n\n Integration of sin x cos x dx can be found using various integration technique like integration by substitution, Integration by partial fraction along with trigonometric identities. The various formula for integration of sin x cos x dx are I \\[\\int \\sin(x) \\cos (x)\\, dx = -\\frac {1}{4} \\cos (2x) + C\\] II \\[\\int \\sin(x) \\cos […]<\/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-8841","post","type-post","status-publish","format-standard","hentry","category-maths"],"yoast_head":"\n
$$
\\int_{0}^{\\pi\/4} \\cos x \\sin x \\, dx = \\frac{1}{2} \\int_{0}^{\\pi\/4} \\sin(2x) \\, dx
$$<\/p>\n\n\n\n
\\frac{1}{2} \\left[ -\\frac{1}{2} \\cos(2x) \\right]_{0}^{\\pi\/4}
$$<\/p>\n\n\n\n
= \\frac{1}{2} \\left( -\\frac{1}{2} \\cos(\\pi\/2) + \\frac{1}{2} \\cos(0) \\right)
$$<\/p>\n\n\n\n
= \\frac{1}{2} \\left( -\\frac{1}{2} \\cdot 0 + \\frac{1}{2} \\cdot 1 \\right)
= \\frac{1}{4}
$$<\/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":"