This page covers the derivation and use of the cosine reduction formula for integration.
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Cosine Reduction Formula (where n is a positive integer) 

\(\displaystyle{\int{\cos^n x~dx} = \frac{\cos^{n1}x\sin x}{n} + \frac{n1}{n}\int{\cos^{n2}x~dx}}\) 
When you have an integral with only cosine where the power is greater than one, you can use the cosine reduction formula, repeatedly if necessary, to reduce the power until you end up with either \(\cos x\) or \(\cos^2 x\). Let's derive the formula and then work some practice problems.
Deriving The Cosine Reduction Formula
Separate out one \(\cos x\) term. 
\(\int{\cos^nx~dx} = \int{\cos^{n1}x \cos x~dx}\) 
Use integration by parts. 
\(u=\cos^{n1}x \to \) \(du=(n1)\cos^{n2}x(\sin x)~dx\) 
\(dv=\cos x~dx \to v=\sin x\) 
\(\int{\cos^nx~dx} =\) \(\sin x\cos^{n1}x\) \(+\) \(\int{(n1)\cos^{n2}x\sin^2x~dx}\) 
Use \(\sin^2x + \cos^2x = 1\) to replace the \(\sin^2x\) in the last integral with \(1\cos^2x\). 
\(\int{\cos^nx~dx} =\) \(\sin x\cos^{n1}x \) \(+\) \((n1)\int{\cos^{n2}x~(1\cos^2x)~dx}\) 
In the last integral, distribute the \(\cos^{n2}x\) term and separate the integral into two integrals. Don't forget to distribute the \((n1)\) term as well. 
\(\int{\cos^nx~dx} =\) \(\sin x\cos^{n1}x \) \(+\) \((n1)\int{\cos^{n2}x~dx} \) \(\) \((n1)\int{\cos^nx~dx}\) 
Now add \((n1)\int{\cos^nx~dx}\) to both sides giving us \(n1+1=n\) on the left. 
\(n\int{\cos^nx~dx} =\) \(\sin x\cos^{n1}x \) \(+\) \((n1)\int{\cos^{n2}x~dx}\) 
Solve for \(\int{\cos^nx~dx}\) by dividing both sides by n. 
\(\displaystyle{ \int{\cos^nx~dx} = \frac{\cos^{n1}x\sin x}{n} + \frac{n1}{n}\int{\cos^{n2}x~dx} }\) 

This last equation is the cosine reduction formula. Now let's work some practice problems.
Practice
Unless otherwise instructed, evaluate these integrals directly, then check your answer using the reduction formula.
\(\int{ \cos^2 x ~dx }\)
Problem Statement
Evaluate \(\int{ \cos^2 x ~dx }\) directly using trig identities, then check your answer using the reduction formula.
Solution
video by The Organic Chemistry Tutor 

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\(\int{ \cos^3 x ~dx }\)
Problem Statement
Evaluate \(\int{ \cos^3 x ~dx }\) directly using trig identities, then check your answer using the reduction formula.
Solution
video by The Organic Chemistry Tutor 

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Really UNDERSTAND Calculus
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 
\(\sin(2\theta)=2\sin\theta\cos\theta\)  \(\cos(2\theta)=\cos^2\theta\sin^2\theta\) 
links 
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)  
\(\displaystyle{\int{\sin^n x~dx} = \frac{\sin^{n1}x\cos x}{n} + }\) \(\displaystyle{ \frac{n1}{n}\int{\sin^{n2}x~dx} }\)  
\(\displaystyle{\int{\cos^n x~dx} = \frac{\cos^{n1}x\sin x}{n} + }\) \(\displaystyle{ \frac{n1}{n}\int{\cos^{n2}x~dx}}\)  
Reduction Formulas (n is an integer and \(n>1\))  
\(\displaystyle{\int{\tan^n x~dx}= \frac{\tan^{n1}x}{n1}  \int{\tan^{n2}x~dx}}\)  
\(\displaystyle{\int{\sec^n x~dx} = \frac{\sec^{n2}x\tan x}{n1} + }\) \(\displaystyle{ \frac{n2}{n1}\int{\sec^{n2}x~dx}}\)  
links  
related topics on other pages 

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Practice Instructions
Unless otherwise instructed, evaluate these integrals directly, then check your answer using the reduction formula.