The integral of the cosec square function, $\\csc ^2x$, can be found using basic calculus principles. The integral of ($\\csc ^2x$) with respect to (x) is:<\/p>\n\n\n\n
\\[
\\int \\csc^2 x \\, dx = -\\cot 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. This result is derived from the fact that the derivative of ($\\cot x$) is $-\\csc^2 x$.<\/p>\n\n\n\n
we can use the fundamental theorem of calculus, which states that if a function f is the derivative of another function F, then the integral of f is F plus a constant.<\/p>\n\n\n\n
Now we know that<\/p>\n\n\n\n
$\\frac {d}{dx} \\cot (x)= – \\csc^2(x)$
$\\frac {d}{dx} (- \\cot(x)) = \\csc^2(x)$
or
$\\frac {d}{dx} (- \\cot(x)+ C) = \\csc^2(x)$<\/p>\n\n\n\n
Where C is the constant<\/p>\n\n\n\n
Now from fundamental theorem of calculus stated above, we can say that<\/p>\n\n\n\n
\\[
\\int \\csc^2 x \\, dx = -\\cot x + C
\\]<\/p>\n\n\n\n
Let $t= \\cot x$
$dt = -\\csc^2 x dx$
Therefore<\/p>\n\n\n\n
\\[
\\int \\csc^2 x \\, dx = – \\int \\; dt
\\]<\/p>\n\n\n\n
$= -t + C$<\/p>\n\n\n\n
Substituting back the values<\/p>\n\n\n\n
\\[
\\int \\csc^2 x \\, dx = -\\cot x + C
\\]<\/p>\n\n\n\n
Example 1<\/strong><\/p>\n\n\n\n $\\int x \\csc^2 x \\, dx$<\/p>\n\n\n\n Solution<\/strong><\/p>\n\n\n\n To solve the integral , we need to use integration by parts, which is given by the formula:<\/p>\n\n\n\n $$ Here, we can choose $ u = x $ and $ dv = \\csc^2 x \\, dx $. Then we need to find $ du $ and $ v $.<\/p>\n\n\n\n The integral of $\\csc^2 x$ is $ -\\cot x $, so $ v = -\\cot x $.<\/p>\n\n\n\n Substituting these<\/p>\n\n\n\n $$ Now we know that $\\int \\cot x \\, dx= \\log|\\sin x|$.<\/p>\n\n\n\n Therefore, the final value is where $C$ is the constant of integration.<\/p>\n\n\n\n Example 2<\/strong><\/p>\n\n\n\n $\\int \\csc^2 (2x) \\, dx$<\/p>\n\n\n\n Solution<\/strong><\/p>\n\n\n\n Let $t= 2 x$ $\\int \\csc^2 (2x) \\, dx = \\frac {1}{2} \\int \\csc^2 (t) \\, dt $ Substituting back<\/p>\n\n\n\n $\\int \\csc^2 (2x) \\, dx = \\frac {-1}{2} ( \\cot 2x) + C)$<\/p>\n\n\n\n I hope you like article on Integration of cosec 2x. <\/p>\n\n\n\n Other Integration Related Articles<\/p>\n\n\n\n The integral of the cosec square function, $\\csc ^2x$, can be found using basic calculus principles. The integral of ($\\csc ^2x$) with respect to (x) is: \\[\\int \\csc^2 x \\, dx = -\\cot x + C\\] Here, (C) represents the constant of integration, which is added because the process of integration determines the antiderivative up […]<\/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-8886","post","type-post","status-publish","format-standard","hentry","category-maths"],"yoast_head":"\n
\\int u \\, dv = uv – \\int v \\, du
$$<\/p>\n\n\n\n\n
\\int x \\csc^2 x \\, dx = x(-\\cot x) – \\int (-\\cot x) \\, dx
$$<\/p>\n\n\n\n
$$
=-x \\cot x + \\log|\\sin x| + C
$$<\/p>\n\n\n\n
$dt = 2 dx$
Therefore<\/p>\n\n\n\n
$= -\\frac {1}{2}( \\cot t + C)$<\/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":"