## 17Calculus Integrals - Cosine Reduction Formula

##### 17Calculus

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

Cosine Reduction Formula (where n is a positive integer)

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

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

$$\displaystyle{ \int{\cos^nx~dx} = \frac{\cos^{n-1}x\sin x}{n} + \frac{n-1}{n}\int{\cos^{n-2}x~dx} }$$ Separate out one $$\cos x$$ term. $$\int{\cos^nx~dx} = \int{\cos^{n-1}x \cos x~dx}$$ Use integration by parts. $$u=\cos^{n-1}x \to$$ $$du=(n-1)\cos^{n-2}x(-\sin x)~dx$$ $$dv=\cos x~dx \to v=\sin x$$ $$\int{\cos^nx~dx} =$$ $$\sin x\cos^{n-1}x$$ $$+$$ $$\int{(n-1)\cos^{n-2}x\sin^2x~dx}$$ Use $$\sin^2x + \cos^2x = 1$$ to replace the $$\sin^2x$$ in the last integral with $$1-\cos^2x$$. $$\int{\cos^nx~dx} =$$ $$\sin x\cos^{n-1}x$$ $$+$$ $$(n-1)\int{\cos^{n-2}x~(1-\cos^2x)~dx}$$ In the last integral, distribute the $$\cos^{n-2}x$$ term and separate the integral into two integrals. Don't forget to distribute the $$(n-1)$$ term as well. $$\int{\cos^nx~dx} =$$ $$\sin x\cos^{n-1}x$$ $$+$$ $$(n-1)\int{\cos^{n-2}x~dx}$$ $$-$$ $$(n-1)\int{\cos^nx~dx}$$ Now add $$(n-1)\int{\cos^nx~dx}$$ to both sides giving us $$n-1+1=n$$ on the left. $$n\int{\cos^nx~dx} =$$ $$\sin x\cos^{n-1}x$$ $$+$$ $$(n-1)\int{\cos^{n-2}x~dx}$$ Solve for $$\int{\cos^nx~dx}$$ by dividing both sides by n.

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

### The Organic Chemistry Tutor - 2578 video solution

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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

### The Organic Chemistry Tutor - 2579 video solution

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### trig integration 17calculus youtube playlist

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