Integration of tan x can be found using various integration technique like integration by substitution along with trigonometric identities. The formula for integration of tan x is<\/p>\n\n\n\n
\\[
\\int \\tan(x) \\, dx = \\ln |sec(x)| + C
\\]<\/p>\n\n\n\n
Integration of tan x can be solved using integration by substitution as given below<\/p>\n\n\n\n
$\\int \\frac {f^{‘} (x)}{f(x)} \\; dx = ln | f(x)| + C$ Now<\/p>\n\n\n\n \\[ Now \\[ \\[ To evaluate the definite integral of $\\tan(x)$ over a specific interval, say from (a) to (b), you follow the same process as for the indefinite integral, but then apply the limits of integration. The indefinite integral of $\\tan(x)$ is $\\ln|\\sec(x)| + C$.<\/p>\n\n\n\n So, the definite integral <\/p>\n\n\n\n \\[ However, it’s important to be cautious about the interval of integration because $\\tan(x)$ and $\\ln|\\sec(x)|$ have singularities (points where they are not defined). Specifically, $\\tan(x)$ and $\\ln|\\sec(x)|$ are undefined where $\\cos(x) = 0$, which occurs $\\frac {\\pi}{2} + k \\pi$ where k is an integers. Therefore, the interval ([a, b]) should not include points where $\\cos(x) = 0$<\/p>\n\n\n\n Example<\/strong><\/p>\n\n\n\n \\[ However, this integral is problematic at ($x =\\pi\/2$) ) because $\\ln|\\sec(\\pi\/2)|$ is undefined (as $\\sec(\\pi\/2)$ is not defined). In such cases, the integral does not have a standard value<\/p>\n\n\n\n Question 1<\/strong><\/p>\n\n\n\n Find<\/p>\n\n\n\n \\[ Solution<\/strong><\/p>\n\n\n\n \\[ Now, evaluate this at the upper and lower limits:<\/p>\n\n\n\n \\[ Other Integration Related Articles<\/p>\n\n\n\n Integration of tan x can be found using various integration technique like integration by substitution along with trigonometric identities. The formula for integration of tan x is \\[\\int \\tan(x) \\, dx = \\ln |sec(x)| + C\\] Proof of the Integration of tan x Integration of tan x can be solved using integration by substitution as […]<\/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-8622","post","type-post","status-publish","format-standard","hentry","category-maths","tag-integration"],"yoast_head":"\n
Proof<\/strong>
let us substitute
f(x) =t
then
f'(x) dx=dt
Therefore
$\\int \\frac {f^{‘} (x)}{f(x)}\\; dx = \\int \\frac {1}{t} \\; dt =ln |t| + C= ln | f(x)| + C $<\/p>\n\n\n\n
\\int \\tan(x) \\, dx = \\int \\frac {\\sin(x)}{\\cos(x)} \\, dx
\\]<\/p>\n\n\n\n
$\\cos(x) =t$
then
$-\\sin (x) dx = dt$
Therefore,<\/p>\n\n\n\n
\\int \\cot(x) \\, dx =- \\int \\frac {1}{t} \\, dt = -\\ln |cos(x)| + C
\\]<\/p>\n\n\n\n
= \\ln |cos(x)|^{-1} + C=\\ln |\\frac {1}{\\cos(x)}| + C= \\ln |sec(x)| + C
\\]<\/p>\n\n\n\nDefinite Integral of tan x<\/h2>\n\n\n\n
\\int_{a}^{b} \\tan(x) \\, dx = \\left[ \\ln|\\sec(x)| \\right]_{a}^{b} = \\ln|\\sec(b)| – \\ln|\\sec(a)|.
\\]<\/p>\n\n\n\n
\\int_{0}^{\\pi\/2} \\tan(x) \\, dx = \\left[ \\ln|\\sec(x)| \\right]_{0}^{\\pi\/2} = \\ln|\\sec(\\pi\/2)| – \\ln|\\sec(0)| = \\ln( undefined) – \\ln(1).
\\]<\/p>\n\n\n\nSolved Examples<\/h2>\n\n\n\n
\\int_{0}^{\\pi\/4} \\tan(x) \\, dx
\\]<\/p>\n\n\n\n
\\int_{0}^{\\pi\/4} \\tan(x) \\, dx = \\left[ \\ln|\\sec(x)| \\right]_{0}^{\\pi\/4}.
\\]<\/p>\n\n\n\n
\\begin{align} \\left[ \\ln|\\sec(x)| \\right]_{0}^{\\pi\/4} = \\ln|\\sec(\\pi\/4)| – \\ln|\\cos(0)| \\\\ = \\ln\\left|\\sqrt{2}\\right| – \\ln|1| \\\\ = \\ln(\\sqrt{2}). \\end{align}
\\]<\/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":"