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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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SineCosine Trig Integration  

on this page: ► one angle ► reduction formulas ► two angles  
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 sines and/or cosines in more advanced form that require techniques other than just integration by substitution. [If you are first learning sine and cosine in integration, check out the basics of trig integration page.]  
Trig Integration  Complete Case List Summary (One Angle)

In order to choose the technique you need to use, you need to determine the form of the integrand and how many angles are involved.
One Angle 

When all the sine and cosine terms in the integrand involve the same angle, here is what you do.
Case 1 ( one sine term only ): \( \int{ \sin^n(\theta) ~d\theta} \) 
Use the reduction formula found in the section below. 
Case 2 ( one cosine term only ): \( \int{ \cos^n(\theta) ~d\theta} \) 
Using the same technique found in the reduction section below, 
Case 3: \( \int{ \sin^m(\theta)~\cos^n(\theta) ~d\theta} \) with odd m or n 
Factor out the odd term (if they are both odd, you can choose) and 
Case 4: \( \int{ \sin^m(\theta)~\cos^n(\theta) ~d\theta} \) with even m and n 
Use the halfangle formulas to remove the powers. 
Reduction Formulas 

When you have an integral with only sine where the power is greater than one, you can use the sine reduction formula, repeatedly if necessary, to reduce the power until you end up with either \(\sin x\) or \(\sin^2 x\). The formula is derived on a separate page. The reduction formula for cosine is derived on this page.
Reduction Formulas (where n is a positive integer)  

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

\(\displaystyle{\int{\cos^n x~dx} = \frac{\cos^{n1}x\sin x}{n} + \frac{n1}{n}\int{\cos^{n2}x~dx}}\) 
Two Angles 

When you have different angles, in addition to the doubleangle formulas at the top of the page, use the following formulas to simplify the integrand.
\(\displaystyle{ \sin(\alpha)\sin(\theta) = \frac{1}{2}\left[ \cos(\alpha  \theta)  \cos(\alpha + \theta) \right] }\) 
\(\displaystyle{ \cos(\alpha)\cos(\theta) = \frac{1}{2}\left[ \cos(\alpha  \theta) + \cos(\alpha + \theta) \right] }\) 
\(\displaystyle{ \sin(\alpha)\cos(\theta) = \frac{1}{2}\left[ \sin(\alpha + \theta) + \sin(\alpha  \theta) \right] }\) 
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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 A02  

\(\displaystyle{\int{\sin(4x)\cos(2x)~dx}}\)  
solution 
Practice A03  

\(\displaystyle{\int{\cos^5x\sin^5x~dx}}\)  
solution 
Practice A04  

\(\displaystyle{\int{\cos^4x\sin^3x~dx}}\)  
solution 
Level B  Intermediate 
Practice B02  

\(\displaystyle{\int_{0}^{\pi/2}{\sin^7\theta~\cos^5\theta~d\theta}}\)  
solution 
Practice B03  

\(\displaystyle{\int{\cos^4x~dx}}\)  
solution 
Level C  Advanced 
Practice C01  

\(\displaystyle{\int_{0}^{\pi/2}{\sin^7x~dx}}\)  
solution 