\( \newcommand{\abs}[1]{\left| \, {#1} \, \right| } \) \( \newcommand{\cm}{\mathrm{cm} } \) \( \newcommand{\sec}{ \, \mathrm{sec} \, } \) \( \newcommand{\vhat}[1]{\,\hat{#1}} \) \( \newcommand{\vhati}{\,\hat{i}} \) \( \newcommand{\vhatj}{\,\hat{j}} \) \( \newcommand{\vhatk}{\,\hat{k}} \) \( \newcommand{\vect}[1]{\boldsymbol{\vec{#1}}} \) \( \newcommand{\norm}[1]{\|{#1}\|} \) \( \newcommand{\arccot}{ \, \mathrm{arccot} \, } \) \( \newcommand{\arcsec}{ \, \mathrm{arcsec} \, } \) \( \newcommand{\arccsc}{ \, \mathrm{arccsc} \, } \) \( \newcommand{\sech}{ \, \mathrm{sech} \, } \) \( \newcommand{\csch}{ \, \mathrm{csch} \, } \) \( \newcommand{\arcsinh}{ \, \mathrm{arcsinh} \, } \) \( \newcommand{\arccosh}{ \, \mathrm{arccosh} \, } \) \( \newcommand{\arctanh}{ \, \mathrm{arctanh} \, } \) \( \newcommand{\arccoth}{ \, \mathrm{arccoth} \, } \) \( \newcommand{\arcsech}{ \, \mathrm{arcsech} \, } \) \( \newcommand{\arccsch}{ \, \mathrm{arccsch} \, } \)

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This page covers integration of functions involving secants and/or tangents in more advanced form that require techniques other than just integration by substitution. [If you are first learning secant and tangent in integration, check out the basics of trig integration page.]

Difference Between Trig Integration and Trig Substitution

Trig integration, covered on this page, is the evaluation of integrals that already have trig functions in the integrand.

Trig substitution is a technique that takes an integrand that most likely does NOT contain any trig functions, and uses some trig identities to introduce trig functions into the integrand. Once the integral is completely transformed, then trig integration is used to evaluate the integral. Once the evaluation is complete, another set of substitutions, based on the original ones, is done to convert the result back to the original variable.

Trig Integration - Complete Case List Summary (One Angle)

Trig Integration - Complete Case List Summary (One Angle)

\(\int{\sin^mx \cos^nx~dx}\)

\(n=0\)

\(\int{\sin^mx~dx}\)

use reduction formula

\(m=0\)

\(\int{\cos^mx~dx}\)

use reduction formula

odd m

\(\int{\sin^{2k+1}x\cos^nx~dx}\)

factor out \(\sin x\), use \(\sin^2x=1-\cos^2x\) and let \(u=\cos x\)

odd n

\(\int{\sin^mx\cos^{2k+1}x~dx}\)

factor out \(\cos x\), use \(\cos^2x=1-\sin^2x\) and let \(u=\sin x\)

even m and n

\(\int{\sin^{2k}x\cos^{2p}x~dx}\)

use half-angle formulas

\(\int{\sec^mx \tan^nx~dx}\)

\(n=0\)

\(\int{\sec^mx~dx}\)

use reduction formula

\(m=0\)

\(\int{\tan^nx~dx}\)

use reduction formula or use \(\sec^2x=1+\tan^2x\), expand out
and try one of the following two cases

even m

\(\int{\sec^{2k}x\tan^nx~dx}\)

factor out \(\sec^2x\), use \(\sec^2x=1+\tan^2x\) and let \(u=\tan x\)

odd n

\(\int{\sec^mx\tan^{2k+1}x~dx}\)

factor out \(\sec x\tan x\), use \(\sec^2x=1+\tan^2x\) and let \(u=\sec x\)

none of the above 4 cases hold

convert trig functions to sine and cosine and
try the sine/cosine techniques

Integrands Involving Secant and/or Tangent

In order to choose the technique you need to use, you need to determine the form of the integrand.

Case 1 - one secant term only: \( \int{ \sec^n(x) ~dx} \)

Use the reduction formula found on the secant reduction formula page.

 

Case 2 - one tangent term only: \( \int{ \tan^n(x) ~dx} \)

Use \(\sec^2x = 1 + \tan^2x\), expand out and see if you can use case 3 and/or case 4 on the terms or use the reduction formula found on the tangent reduction formula page

 

Case 3 - even secant term: \( \int{ \sec^m(x)~\tan^n(x) ~dx} \) with even m

Factor out \(\sec^2(x)\), use \(\sec^2x = 1 + \tan^2x\) and let \(u = \tan(x)\).

 

Case 4 - odd tangent term: \( \int{ \sec^m(x)~\tan^n(x) ~dx} \) with odd n

Factor out \(\sec(x)\tan(x)\), use \(\sec^2x = 1 + \tan^2x\) and let \(u = \sec(x)\).

Practice

Unless otherwise instructed, evaluate these integrals using the techniques on this page. Give all answers in exact, simplified form.

\(\displaystyle{ \int{ \sec^4x ~\tan^6x ~dx } }\)

Problem Statement

Evaluate \(\displaystyle{ \int{ \sec^4x ~\tan^6x ~dx } }\). Give your answer in exact, simplified and factored form.

Solution

103 video

video by PatrickJMT

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\(\displaystyle{ \int{ \frac{1}{\cos^6(x) \cot^2(x)} ~dx } }\)

Problem Statement

Evaluate \(\displaystyle{ \int{ \frac{1}{\cos^6(x) \cot^2(x)} ~dx } }\). Give your answer in exact, simplified and factored form.

Solution

104 video

video by PatrickJMT

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\(\displaystyle{ \int{ \tan^3(x) ~\sec(x) ~dx } }\)

Problem Statement

Evaluate \(\displaystyle{ \int{ \tan^3(x) ~\sec(x) ~dx } }\). Give your answer in exact, simplified and factored form.

Solution

388 video

video by Krista King Math

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\(\displaystyle{ \int{ \tan^4x ~\sec^6x ~dx } }\)

Problem Statement

Evaluate \(\displaystyle{ \int{ \tan^4x ~\sec^6x ~dx } }\). Give your answer in exact, simplified and factored form.

Solution

398 video

video by Krista King Math

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\(\displaystyle{ \int{ \sec^4x ~\tan^2x ~dx } }\)

Problem Statement

Evaluate \(\displaystyle{ \int{ \sec^4x ~\tan^2x ~dx } }\). Give your answer in exact, simplified and factored form.

Solution

95 video

video by PatrickJMT

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\(\displaystyle{ \int{ \sec^3x ~\tan^5x ~dx } }\)

Problem Statement

Evaluate \(\displaystyle{ \int{ \sec^3x ~\tan^5x ~dx } }\). Give your answer in exact, simplified and factored form.

Solution

97 video

video by PatrickJMT

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\(\displaystyle{ \int{ \tan(x) ~dx } }\)

Problem Statement

Evaluate \(\displaystyle{ \int{ \tan(x) ~dx } }\). Give your answer in exact, simplified, factored form.

Solution

This problem is solved by two different instructors in these videos.

98 video

video by PatrickJMT

98 video

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\(\displaystyle{ \int{ \sec(x) \tan(x) ~dx } }\)

Problem Statement

Evaluate \(\displaystyle{ \int{ \sec(x) \tan(x) ~dx } }\). Give your answer in exact, simplified and factored form.

Solution

91 video

video by Krista King Math

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\(\displaystyle{ \int{ \tan^2x ~dx } }\)

Problem Statement

Evaluate \(\displaystyle{ \int{ \tan^2x ~dx } }\). Give your answer in exact, simplified and factored form.

Solution

2575 video

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\(\displaystyle{ \int{ \sec(x) ~dx } }\)

Problem Statement

Evaluate \(\displaystyle{ \int{ \sec(x) ~dx } }\). Give your answer in exact, simplified and factored form.

Solution

Here are two videos, by two different instructors, solving this problem.

121 video

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You CAN Ace Calculus

Topics You Need To Understand For This Page

Trig Identities and Formulas

basic trig identities

\(\sin^2\theta+\cos^2\theta=1\)   |   \(1+\tan^2\theta=\sec^2\theta\)

\(\displaystyle{\tan\theta=\frac{\sin\theta}{\cos\theta}}\)   |   \(\displaystyle{\cot\theta=\frac{\cos\theta}{\sin\theta}}\)

\(\displaystyle{\sec\theta=\frac{1}{\cos\theta}}\)   |   \(\displaystyle{\csc\theta=\frac{1}{\sin\theta}}\)

power reduction (half-angle) formulae

\(\displaystyle{\sin^2\theta=\frac{1-\cos(2\theta)}{2}}\)   |   \(\displaystyle{\cos^2\theta=\frac{1+\cos(2\theta)}{2}}\)

double angle formulae

\(\sin(2\theta)=2\sin\theta\cos\theta\)   |   \(\cos(2\theta)=\cos^2\theta-\sin^2\theta\)

links

list of trigonometric identities - wikipedia

trig sheets - pauls online notes

17calculus trig formulas - full list

Trig Derivatives and Integrals

basic trig derivatives

\(\displaystyle{ \frac{d[\sin(t)]}{dt} = \cos(t) }\)

\(\displaystyle{ \frac{d[\cos(t)]}{dt} = -\sin(t) }\)

\(\displaystyle{ \frac{d[\tan(t)]}{dt} = \sec^2(t) }\)

\(\displaystyle{ \frac{d[\cot(t)]}{dt} = -\csc^2(t) }\)

\(\displaystyle{ \frac{d[\sec(t)]}{dt} = }\) \(\sec(t)\tan(t) \)

\(\displaystyle{ \frac{d[\csc(t)]}{dt} = }\) \( -\csc(t)\cot(t) \)

basic trig integrals

\(\int{\sin(x)~dx} = -\cos(x)+C\)

\(\int{\cos(x)~dx} = \sin(x)+C\)

\(\int{\tan(x)~dx} = -\ln\abs{\cos(x)}+C\)

\(\int{\cot(x)~dx} = \ln\abs{\sin(x)}+C\)

\(\int{\sec(x)~dx} = \ln\abs{\sec(x)+\tan(x)}+C\)

\(\int{\csc(x)~dx} = -\ln\abs{\csc(x)+\cot(x)}+C\)

reduction formulae

Reduction Formulas (n is a positive integer)

\(\displaystyle{\int{\sin^n x~dx} = -\frac{\sin^{n-1}x\cos x}{n} + }\) \(\displaystyle{ \frac{n-1}{n}\int{\sin^{n-2}x~dx} }\)

\(\displaystyle{\int{\cos^n x~dx} = \frac{\cos^{n-1}x\sin x}{n} + }\) \(\displaystyle{ \frac{n-1}{n}\int{\cos^{n-2}x~dx}}\)

Reduction Formulas (n is an integer and \(n>1\))

\(\displaystyle{\int{\tan^n x~dx}= \frac{\tan^{n-1}x}{n-1} - \int{\tan^{n-2}x~dx}}\)

\(\displaystyle{\int{\sec^n x~dx} = \frac{\sec^{n-2}x\tan x}{n-1} + }\) \(\displaystyle{ \frac{n-2}{n-1}\int{\sec^{n-2}x~dx}}\)

links

17calculus trig formulas - full list

Related Topics and Links

related topics on other pages

basic trig integration

sine-cosine trig integration

external links you may find helpful

Wikipedia - List of Trig Identities

Trig Formulas

The Unit Circle

The Unit Circle [wikipedia]

Basic Trig Identities

Set 1 - basic identities

\(\displaystyle{ \tan(t) = \frac{\sin(t)}{\cos(t)} }\)

\(\displaystyle{ \cot(t) = \frac{\cos(t)}{\sin(t)} }\)

\(\displaystyle{ \sec(t) = \frac{1}{\cos(t)} }\)

\(\displaystyle{ \csc(t) = \frac{1}{\sin(t)} }\)

Set 2 - squared identities

\( \sin^2t + \cos^2t = 1\)

\( 1 + \tan^2t = \sec^2t\)

\( 1 + \cot^2t = \csc^2t\)

Set 3 - double-angle formulas

\( \sin(2t) = 2\sin(t)\cos(t)\)

\(\displaystyle{ \cos(2t) = \cos^2(t) - \sin^2(t) }\)

Set 4 - half-angle formulas

\(\displaystyle{ \sin^2(t) = \frac{1-\cos(2t)}{2} }\)

\(\displaystyle{ \cos^2(t) = \frac{1+\cos(2t)}{2} }\)

Trig Derivatives

\(\displaystyle{ \frac{d[\sin(t)]}{dt} = \cos(t) }\)

 

\(\displaystyle{ \frac{d[\cos(t)]}{dt} = -\sin(t) }\)

\(\displaystyle{ \frac{d[\tan(t)]}{dt} = \sec^2(t) }\)

 

\(\displaystyle{ \frac{d[\cot(t)]}{dt} = -\csc^2(t) }\)

\(\displaystyle{ \frac{d[\sec(t)]}{dt} = \sec(t)\tan(t) }\)

 

\(\displaystyle{ \frac{d[\csc(t)]}{dt} = -\csc(t)\cot(t) }\)

Inverse Trig Derivatives

\(\displaystyle{ \frac{d[\arcsin(t)]}{dt} = \frac{1}{\sqrt{1-t^2}} }\)

 

\(\displaystyle{ \frac{d[\arccos(t)]}{dt} = -\frac{1}{\sqrt{1-t^2}} }\)

\(\displaystyle{ \frac{d[\arctan(t)]}{dt} = \frac{1}{1+t^2} }\)

 

\(\displaystyle{ \frac{d[\arccot(t)]}{dt} = -\frac{1}{1+t^2} }\)

\(\displaystyle{ \frac{d[\arcsec(t)]}{dt} = \frac{1}{\abs{t}\sqrt{t^2 -1}} }\)

 

\(\displaystyle{ \frac{d[\arccsc(t)]}{dt} = -\frac{1}{\abs{t}\sqrt{t^2 -1}} }\)

Trig Integrals

\(\int{\sin(x)~dx} = -\cos(x)+C\)

 

\(\int{\cos(x)~dx} = \sin(x)+C\)

\(\int{\tan(x)~dx} = -\ln\abs{\cos(x)}+C\)

 

\(\int{\cot(x)~dx} = \ln\abs{\sin(x)}+C\)

\(\int{\sec(x)~dx} = \) \( \ln\abs{\sec(x)+\tan(x)}+C\)

 

\(\int{\csc(x)~dx} = \) \( -\ln\abs{\csc(x)+\cot(x)}+C\)

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Unless otherwise instructed, evaluate these integrals using the techniques on this page. Give all answers in exact, simplified form.

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