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17Calculus Integrals - Cosine Reduction Formula

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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

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.

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

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

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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

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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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