Integrating hyperbolic functions is similar to integrating their trigonometric counterparts, but there are some key differences due to the properties of hyperbolic functions. Here are the basic integrals for the six primary hyperbolic functions:<\/p>\n\n\n\n
(1) Sinh (Hyperbolic Sine):<\/strong> Proof<\/strong> $$ $$ $$ $$ And this is the hyperbolic cosine function,$\\cosh(x)). Therefore:<\/p>\n\n\n\n $$ where$C) is the constant of integration.<\/p>\n\n\n\n (2) Cosh (Hyperbolic Cosine):<\/strong> Proof<\/strong> $$ $$ $$ $$ And this is the hyperbolic sin function,$\\sinh(x)). Therefore:<\/p>\n\n\n\n $$ where$C) is the constant of integration.<\/p>\n\n\n\n (3) Tanh (Hyperbolic Tangent):<\/strong> Proof<\/strong> $$ Let t= cosh(x) $$ And this is the hyperbolic tan function,$\\tanh(x)). Therefore:<\/p>\n\n\n\n $$ where$C) is the constant of integration.<\/p>\n\n\n\n (4) Coth (Hyperbolic Cotangent):<\/strong> Proof<\/strong> $$ Let t= sinh(x) $$ And this is the hyperbolic cot function,$\\coth(x)). Therefore:<\/p>\n\n\n\n $$ where$C) is the constant of integration.<\/p>\n\n\n\n (5) Sech (Hyperbolic Secant):<\/strong> (6) Csch (Hyperbolic Cosecant):<\/strong> Other Integration Related Articles<\/p>\n\n\n\n Integrating hyperbolic functions is similar to integrating their trigonometric counterparts, but there are some key differences due to the properties of hyperbolic functions. Here are the basic integrals for the six primary hyperbolic functions: (1) Sinh (Hyperbolic Sine):$$\\int \\sinh(x) \\, dx = \\cosh(x) + C$$The integral of hyperbolic sine is hyperbolic cosine. ProofTo integrate$\\sinh(x)), we […]<\/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":[506],"class_list":["post-8643","post","type-post","status-publish","format-standard","hentry","category-maths","tag-integration"],"yoast_head":"\n
$$
\\int \\sinh(x) \\, dx = \\cosh(x) + C
$$
The integral of hyperbolic sine is hyperbolic cosine.<\/p>\n\n\n\n
To integrate$\\sinh(x)), we integrate this expression:<\/p>\n\n\n\n
\\int \\sinh(x) \\, dx = \\int \\frac{e^x – e^{-x}}{2} \\, dx
$$<\/p>\n\n\n\n
\\int \\sinh(x) \\, dx = \\frac{1}{2} \\int e^x \\, dx – \\frac{1}{2} \\int e^{-x} \\, dx
$$<\/p>\n\n\n\n
\\int \\sinh(x) \\, dx = \\frac{1}{2} (e^x – (-e^{-x})) + C
$$<\/p>\n\n\n\n
\\int \\sinh(x) \\, dx = \\frac{1}{2} (e^x + e^{-x}) + C
$$<\/p>\n\n\n\n
\\int \\sinh(x) \\, dx = \\cosh(x) + C
$$<\/p>\n\n\n\n
$$
\\int \\cosh(x) \\, dx = \\sinh(x) + C
$$
The integral of hyperbolic cosine is hyperbolic sine.<\/p>\n\n\n\n
To integrate$\\sinh(x)), we integrate this expression:<\/p>\n\n\n\n
\\int \\cosh(x) \\, dx = \\int \\frac{e^x + e^{-x}}{2} \\, dx
$$<\/p>\n\n\n\n
\\int \\cosh(x) \\, dx = \\frac{1}{2} \\int e^x \\, dx + \\frac{1}{2} \\int e^{-x} \\, dx
$$<\/p>\n\n\n\n
\\int \\cosh(x) \\, dx = \\frac{1}{2} (e^x – e^{-x}) + C
$$<\/p>\n\n\n\n
\\int \\cosh(x) \\, dx = \\frac{1}{2} (e^x – e^{-x}) + C
$$<\/p>\n\n\n\n
\\int \\cosh(x) \\, dx = \\sinh(x) + C
$$<\/p>\n\n\n\n
$$
\\int \\tanh(x) \\, dx = \\ln(\\cosh(x)) + C
$$
The integral of hyperbolic tangent involves the natural logarithm of hyperbolic cosine.<\/p>\n\n\n\n
To integrate$\\tanh(x)), we integrate this expression:<\/p>\n\n\n\n
\\int \\tanh(x) \\, dx = \\int \\frac{sinh(x)}{cosh(x)} \\, dx
$$<\/p>\n\n\n\n
dt= sinh(x) dx<\/p>\n\n\n\n
\\int \\tanh(x) \\, dx = \\int \\frac {1}{t}\\, dt= ln |t| + C
$$<\/p>\n\n\n\n
\\int \\tan h(x) \\, dx = ln | cosh(x)| + C
$$<\/p>\n\n\n\n
$$
\\int \\coth(x) \\, dx = \\ln|\\sinh(x)| + C
$$
The integral of hyperbolic cotangent involves the natural logarithm of the absolute value of hyperbolic sine.<\/p>\n\n\n\n
To integrate$\\coth(x)), we integrate this expression:<\/p>\n\n\n\n
\\int \\coth(x) \\, dx = \\int \\frac{cosh(x)}{sinh(x)} \\, dx
$$<\/p>\n\n\n\n
dt= cosh(x) dx<\/p>\n\n\n\n
\\int \\coth(x) \\, dx = \\int \\frac {1}{t}\\, dt= ln |t| + C
$$<\/p>\n\n\n\n
\\int \\cot h(x) \\, dx = ln | sinh(x)| + C
$$<\/p>\n\n\n\n
$$
\\int sech(x) \\, dx = \\arctan(\\sinh(x)) + C
$$
The integral of hyperbolic secant involves the inverse tangent of hyperbolic sine.<\/p>\n\n\n\n
$$
\\int csch(x) \\, dx = \\ln|\\tanh(x\/2)| + C
$$
The integral of hyperbolic cosecant involves the natural logarithm of the absolute value of the hyperbolic tangent of half the angle.<\/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":"