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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.

Schaum's Outline of Calculus, 6e: 1,105 Solved Problems + 30 Videos

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

The Organic Chemistry Tutor - 2580 video solution

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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

The Organic Chemistry Tutor - 2577 video solution

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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

The Organic Chemistry Tutor - 2576 video solution

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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.

Michael Penn - 2300 video solution

video by Michael Penn

The Organic Chemistry Tutor - 2300 video solution

PatrickJMT - 2300 video solution

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

Dr Chris Tisdell - 123 video solution

video by Dr Chris Tisdell

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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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