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Topics You Need To Understand For This Page
Trig Identities and Formulae  NEW
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 (halfangle) formulae 
\(\displaystyle{\sin^2\theta=\frac{1\cos(2\theta)}{2}}\)  \(\displaystyle{\cos^2\theta=\frac{1+\cos(2\theta)}{2}}\) 
double angle formulae 
\(\sin(2\theta)=2\sin\theta\cos\theta\)  \(\cos(2\theta)=\cos^2\theta\sin^2\theta\) 
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Trig Derivatives and Integrals  NEW
basic trig derivatives  

\(\displaystyle{ \frac{d[\sin(t)]}{dt} = \cos(t) }\) 
\(\displaystyle{ \frac{d[\cos(t)]}{dt} = \sin(t) }\)  
\(\displaystyle{ \frac{d[\tan(t)]}{dt} = \sec^2(t) }\) 
\(\displaystyle{ \frac{d[\cot(t)]}{dt} = \csc^2(t) }\)  
\(\displaystyle{ \frac{d[\sec(t)]}{dt} = \sec(t)\tan(t) }\) 
\(\displaystyle{ \frac{d[\csc(t)]}{dt} = \csc(t)\cot(t) }\)  
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 (where n is a positive integer)  
\(\displaystyle{\int{\sin^n x~dx} = \frac{\sin^{n1}x\cos x}{n}+\frac{n1}{n}\int{\sin^{n2}x~dx}}\)  
\(\displaystyle{\int{\cos^n x~dx} = \frac{\cos^{n1}x\sin x}{n} + \frac{n1}{n}\int{\cos^{n2}x~dx}}\)  
Reduction Formulas (where n is an integer and \(n>1\))  
\(\displaystyle{\int{\tan^n x~dx}= \frac{\tan^{n1}x}{n1}  \int{\tan^{n2}x~dx}}\)  
\(\displaystyle{\int{\sec^n x~dx} = \frac{\sec^{n2}x\tan x}{n1}+\frac{n2}{n1}\int{\sec^{n2}x~dx}}\)  
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SecantTangent Trig Integration  

on this page: ► integrands involving secant and/or tangent ► reduction formulas  
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.
 
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  Complete Case List Summary (One Angle)

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 in the section below. 
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. 
Case 3: \( \int{ \sec^m(x)~\tan^n(x) ~dx} \) with even m 
Factor out \(\sec^2(x)\), use \(\sec^2x = 1 + \tan^2x\) and let \(u = \tan(x)\). 
Case 4: \( \int{ \sec^m(x)~\tan^n(x) ~dx} \) with odd n 
Factor out \(\sec(x)\tan(x)\), use \(\sec^2x = 1 + \tan^2x\) and let \(u = \sec(x)\). 
Reduction Formulas 

Having reduction formulas can save time when evaluating some integrals. These videos derive the reduction formula for a couple of common integrands.
PatrickJMT  \(\sec^n(x)\)  
In this next video, he uses the reduction formula for \(\tan^n(x)\) to calculate an integral involving \(\tan^5(x)\).
Dr Chris Tisdell  \(\tan^n(x)\)  
Reduction Formulas (where n is an integer and \(n>1\))  

\(\displaystyle{\int{\tan^n x~dx}= \frac{\tan^{n1}x}{n1}  \int{\tan^{n2}x~dx}}\) 

\(\displaystyle{\int{\sec^n x~dx} = \frac{\sec^{n2}x\tan x}{n1}+\frac{n2}{n1}\int{\sec^{n2}x~dx}}\) 
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Practice Problems 

Instructions   Unless otherwise instructed, evaluate the following integrals using the techniques on this page. Give all answers in exact, simplified form.
Here are a few practice problems. You can find more practice problems on this page. 
Level A  Basic 
Practice A01  

\(\displaystyle{\int{\sec^4x~\tan^6x~dx}}\)  
solution 
Practice A02  

\(\displaystyle{\int{\frac{1}{\cos^6(x)\cot^2(x)}~dx}}\)  
solution 
Practice A03  

\(\displaystyle{\int{\tan^3(x)~\sec(x)~dx}}\)  
solution 
Practice A04  

\(\displaystyle{\int{\tan^4x~\sec^6x~dx}}\)  
solution 
Level B  Intermediate 
Practice B01  

\(\displaystyle{\int{\tan^5x~dx}}\)  
solution 
Practice B02  

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