This page covers the derivation and use of the sine reduction formula for integration.
Topics You Need To Understand For This Page 

basic trig integration integration by substitution integration by parts 
Recommended Books on Amazon (affiliate links)  

Sine Reduction Formula (where n is a positive integer) 

\(\displaystyle{\int{\sin^n x~dx} = \frac{\sin^{n1}x\cos x}{n}+\frac{n1}{n}\int{\sin^{n2}x~dx}}\) 
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 \(\sin^2 x\). Let's derive the formula and then work some practice problems.
Deriving The Sine Reduction Formula
First, we separate out one \(\sin(x)\) term. 
\( \int{\sin^nx~dx} = \int{\sin^{n1}x\sin x~dx} \) 
Now we use integration by parts 
\(u=\sin^{n1}x ~~~ \to \) \( ~~~ du=(n1)\sin^{n2}x\cos x~dx\) 
\(dv=\sin x~dx ~~~ \to \) \( ~~~ v = \cos x\) 
\(\int{\sin^nx~dx} = \) \(\cos x\sin^{n1}x  \) \(\int{\cos x(n1)\sin^{n2}x\cos x~dx}\) 
Simplify 
\(\int{\sin^nx~dx} = \) \(\cos x\sin^{n1}x \) + \((n1)\int{\sin^{n2}x\cos^2x~dx}\) 
Now we use the identity \(\cos^2x+\sin^2x=1\) to replace \(\cos^2x\) with \(1\sin^2x\) in the last integral. 
\(\int{\sin^nx~dx} = \) \(\cos x\sin^{n1}x \) + \((n1)\int{\sin^{n2}x~(1\sin^2x)~dx}\) 
Next distribute the \(\sin^{n2}x\). 
\(\int{\sin^nx~dx} = \) \(\cos x\sin^{n1}x \) + \((n1)\int{\sin^{n2}x\sin^nx~dx}\) 
Separate the integral on the right into two integrals. Don't forget that the \((n1)\) factor needs to be applied to both integrals. 
\(\int{\sin^nx~dx} = \) \(\cos x\sin^{n1}x \) + \((n1)\int{\sin^{n2}x~dx}  \) \((n1)\int{\sin^nx~dx}\) 
Now we add \((n1)\int{\sin^nx~dx} \) to both sides of the equal sign. 
\(\int{\sin^nx~dx} + (n1)\int{\sin^nx~dx} = \) \(\cos x\sin^{n1}x \) + \((n1)\int{\sin^{n2}x~dx} \) 
Factor \(\int{\sin^nx~dx}\) on the left and notice that \(1+n1 = n\). 
\(n\int{\sin^nx~dx} = \) \(\cos x\sin^{n1}x \) + \((n1)\int{\sin^{n2}x~dx} \) 
Divide both sides by \(n\) to solve for \(\int{\sin^nx~dx}\). 
\(\displaystyle{\int{\sin^nx~dx} = }\) \(\displaystyle{\frac{\cos x\sin^{n1}x}{n} }\) + \(\displaystyle{\frac{n1}{n}\int{\sin^{n2}x~dx} }\) 
This last equation is the sine reduction formula. Here are a couple of videos showing this derivation in similar ways.
video by Dr Chris Tisdell 

video by Michael Penn 

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

Log in to rate this practice problem and to see it's current rating. 

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

Log in to rate this practice problem and to see it's current rating. 

\(\int{ \sin^4 x ~dx }\)
Problem Statement
Integrate \(\int{ \sin^4 x ~dx }\) directly, then check your answer using the reduction formula.
Solution
video by The Organic Chemistry Tutor 

Log in to rate this practice problem and to see it's current rating. 

\(\int{ \sin^5 x ~dx }\)
Problem Statement
Integrate \(\int{ \sin^5 x ~dx }\) using usubstitution and trig identities, then check your answer using the reduction formula.
Solution
Here are three videos, by three different instructors, solving this problem.
video by Michael Penn 

video by The Organic Chemistry Tutor 

video by PatrickJMT 

Log in to rate this practice problem and to see it's current rating. 

\(\displaystyle{\int_{0}^{\pi/2}{\sin^7x~dx}}\)
Problem Statement
Evaluate \(\displaystyle{\int_{0}^{\pi/2}{\sin^7x~dx}}\) directly, then check your answer using the reduction formula.
Solution
video by Dr Chris Tisdell 

Log in to rate this practice problem and to see it's current rating. 

Really UNDERSTAND Calculus
Log in to rate this page and to see it's current rating.
To bookmark this page and practice problems, log in to your account or set up a free account.
Do you have a practice problem number but do not know on which page it is found? If so, enter the number below and click 'page' to go to the page on which it is found or click 'practice' to be taken to the practice problem.
 
I recently started a Patreon account to help defray the expenses associated with this site. To keep this site free, please consider supporting me. 

Support 17Calculus on Patreon 

