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.]
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) | ||
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\(\int{\sin^mx \cos^nx~dx}\) | ||
\(n=0\) | \(\int{\sin^mx~dx}\) | 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 |
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 |
Recommended Books on Amazon (affiliate links) | ||
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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
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
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
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
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
Here are two solutions by two different instructors.
video by Michael Penn |
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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
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.
video by PatrickJMT |
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video by The Organic Chemistry Tutor |
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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
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
video by The Organic Chemistry Tutor |
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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.
video by The Organic Chemistry Tutor |
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Really UNDERSTAND Calculus
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basic trig identities |
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\(\sin^2\theta+\cos^2\theta=1\) | \(1+\tan^2\theta=\sec^2\theta\) |
\(\displaystyle{\tan\theta=\frac{\sin\theta}{\cos\theta}}\) | \(\displaystyle{\cot\theta=\frac{\cos\theta}{\sin\theta}}\) |
\(\displaystyle{\sec\theta=\frac{1}{\cos\theta}}\) | \(\displaystyle{\csc\theta=\frac{1}{\sin\theta}}\) |
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 |
basic trig derivatives | ||
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\(\displaystyle{ \frac{d[\sin(t)]}{dt} = \cos(t) }\) |
\(\displaystyle{ \frac{d[\cos(t)]}{dt} = -\sin(t) }\) | |
\(\displaystyle{ \frac{d[\tan(t)]}{dt} = \sec^2(t) }\) |
\(\displaystyle{ \frac{d[\cot(t)]}{dt} = -\csc^2(t) }\) | |
\(\displaystyle{ \frac{d[\sec(t)]}{dt} = }\) \(\sec(t)\tan(t) \) |
\(\displaystyle{ \frac{d[\csc(t)]}{dt} = }\) \( -\csc(t)\cot(t) \) | |
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 | ||
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