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17Calculus Integrals - Sine Reduction Formula

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This page covers the derivation and use of the sine reduction formula for integration.

Sine Reduction Formula (where n is a positive integer)

\(\displaystyle{\int{\sin^n x~dx} = -\frac{\sin^{n-1}x\cos x}{n}+\frac{n-1}{n}\int{\sin^{n-2}x~dx}}\)

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.

\( \int{\sin^nx~dx} = \int{\sin^{n-1}x\sin x~dx} \)

Now we use integration by parts

\(u=\sin^{n-1}x ~~~ \to \) \( ~~~ du=(n-1)\sin^{n-2}x\cos x~dx\)

\(dv=\sin x~dx ~~~ \to \) \( ~~~ v = -\cos x\)

\(\int{\sin^nx~dx} = \) \(-\cos x\sin^{n-1}x - \) \(\int{-\cos x(n-1)\sin^{n-2}x\cos x~dx}\)

Simplify

\(\int{\sin^nx~dx} = \) \(-\cos x\sin^{n-1}x \) + \((n-1)\int{\sin^{n-2}x\cos^2x~dx}\)

Now we use the identity \(\cos^2x+\sin^2x=1\) to replace \(\cos^2x\) with \(1-\sin^2x\) in the last integral.

\(\int{\sin^nx~dx} = \) \(-\cos x\sin^{n-1}x \) + \((n-1)\int{\sin^{n-2}x~(1-\sin^2x)~dx}\)

Next distribute the \(\sin^{n-2}x\).

\(\int{\sin^nx~dx} = \) \(-\cos x\sin^{n-1}x \) + \((n-1)\int{\sin^{n-2}x-\sin^nx~dx}\)

Separate the integral on the right into two integrals. Don't forget that the \((n-1)\) factor needs to be applied to both integrals.

\(\int{\sin^nx~dx} = \) \(-\cos x\sin^{n-1}x \) + \((n-1)\int{\sin^{n-2}x~dx} - \) \((n-1)\int{\sin^nx~dx}\)

Now we add \((n-1)\int{\sin^nx~dx} \) to both sides of the equal sign.

\(\int{\sin^nx~dx} + (n-1)\int{\sin^nx~dx} = \) \(-\cos x\sin^{n-1}x \) + \((n-1)\int{\sin^{n-2}x~dx} \)

Factor \(\int{\sin^nx~dx}\) on the left and notice that \(1+n-1 = n\).

\(n\int{\sin^nx~dx} = \) \(-\cos x\sin^{n-1}x \) + \((n-1)\int{\sin^{n-2}x~dx} \)

Divide both sides by \(n\) to solve for \(\int{\sin^nx~dx}\).

\(\displaystyle{\int{\sin^nx~dx} = }\) \(\displaystyle{\frac{-\cos x\sin^{n-1}x}{n} }\) + \(\displaystyle{\frac{n-1}{n}\int{\sin^{n-2}x~dx} }\)

This last equation is the sine reduction formula. Here are a couple of videos showing this derivation in similar ways.

Dr Chris Tisdell - Reduction formula \(\sin^n x\) [5min-4secs]

video by Dr Chris Tisdell

Michael Penn - Integration Example: A power reducing formula for sin(x)

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

2580 video

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\(\int{ \sin^3 x ~dx }\)

Problem Statement

Integrate \(\int{ \sin^3 x ~dx }\) directly, then check your answer using the reduction formula.

Solution

2577 video

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\(\int{ \sin^4 x ~dx }\)

Problem Statement

Integrate \(\int{ \sin^4 x ~dx }\) directly, then check your answer using the reduction formula.

Solution

2576 video

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\(\int{ \sin^5 x ~dx }\)

Problem Statement

Integrate \(\int{ \sin^5 x ~dx }\) using u-substitution and trig identities, then check your answer using the reduction formula.

Solution

Here are three videos, by three different instructors, solving this problem.

2300 video

video by Michael Penn

2300 video

2300 video

video by PatrickJMT

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

Problem Statement

Evaluate \(\displaystyle{\int_{0}^{\pi/2}{\sin^7x~dx}}\) directly, then check your answer using the reduction formula.

Solution

123 video

video by Dr Chris Tisdell

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trig integration 17calculus youtube playlist

Really UNDERSTAND Calculus

Topics You Need To Understand For This Page

Trig Identities and Formulas

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 (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

list of trigonometric identities - wikipedia

trig sheets - pauls online notes

17calculus trig formulas - full list

Trig Derivatives and Integrals

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)

\(\displaystyle{\int{\sin^n x~dx} = -\frac{\sin^{n-1}x\cos x}{n} + }\) \(\displaystyle{ \frac{n-1}{n}\int{\sin^{n-2}x~dx} }\)

\(\displaystyle{\int{\cos^n x~dx} = \frac{\cos^{n-1}x\sin x}{n} + }\) \(\displaystyle{ \frac{n-1}{n}\int{\cos^{n-2}x~dx}}\)

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

17calculus trig formulas - full list

Related Topics and Links

related topics on other pages

basic trig integration

secant-tangent trig integration

external links you may find helpful

Wikipedia - List of Trig Identities

Trig Formulas

The Unit Circle

The Unit Circle [wikipedia]

Basic Trig Identities

Set 1 - basic identities

\(\displaystyle{ \tan(t) = \frac{\sin(t)}{\cos(t)} }\)

\(\displaystyle{ \cot(t) = \frac{\cos(t)}{\sin(t)} }\)

\(\displaystyle{ \sec(t) = \frac{1}{\cos(t)} }\)

\(\displaystyle{ \csc(t) = \frac{1}{\sin(t)} }\)

Set 2 - squared identities

\( \sin^2t + \cos^2t = 1\)

\( 1 + \tan^2t = \sec^2t\)

\( 1 + \cot^2t = \csc^2t\)

Set 3 - double-angle formulas

\( \sin(2t) = 2\sin(t)\cos(t)\)

\(\displaystyle{ \cos(2t) = \cos^2(t) - \sin^2(t) }\)

Set 4 - half-angle formulas

\(\displaystyle{ \sin^2(t) = \frac{1-\cos(2t)}{2} }\)

\(\displaystyle{ \cos^2(t) = \frac{1+\cos(2t)}{2} }\)

Trig Derivatives

\(\displaystyle{ \frac{d[\sin(t)]}{dt} = \cos(t) }\)

 

\(\displaystyle{ \frac{d[\cos(t)]}{dt} = -\sin(t) }\)

\(\displaystyle{ \frac{d[\tan(t)]}{dt} = \sec^2(t) }\)

 

\(\displaystyle{ \frac{d[\cot(t)]}{dt} = -\csc^2(t) }\)

\(\displaystyle{ \frac{d[\sec(t)]}{dt} = \sec(t)\tan(t) }\)

 

\(\displaystyle{ \frac{d[\csc(t)]}{dt} = -\csc(t)\cot(t) }\)

Inverse Trig Derivatives

\(\displaystyle{ \frac{d[\arcsin(t)]}{dt} = \frac{1}{\sqrt{1-t^2}} }\)

 

\(\displaystyle{ \frac{d[\arccos(t)]}{dt} = -\frac{1}{\sqrt{1-t^2}} }\)

\(\displaystyle{ \frac{d[\arctan(t)]}{dt} = \frac{1}{1+t^2} }\)

 

\(\displaystyle{ \frac{d[\arccot(t)]}{dt} = -\frac{1}{1+t^2} }\)

\(\displaystyle{ \frac{d[\arcsec(t)]}{dt} = \frac{1}{\abs{t}\sqrt{t^2 -1}} }\)

 

\(\displaystyle{ \frac{d[\arccsc(t)]}{dt} = -\frac{1}{\abs{t}\sqrt{t^2 -1}} }\)

Trig Integrals

\(\int{\sin(x)~dx} = -\cos(x)+C\)

 

\(\int{\cos(x)~dx} = \sin(x)+C\)

\(\int{\tan(x)~dx} = -\ln\abs{\cos(x)}+C\)

 

\(\int{\cot(x)~dx} = \ln\abs{\sin(x)}+C\)

\(\int{\sec(x)~dx} = \) \( \ln\abs{\sec(x)+\tan(x)}+C\)

 

\(\int{\csc(x)~dx} = \) \( -\ln\abs{\csc(x)+\cot(x)}+C\)

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Practice Instructions

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

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