17Calculus Integrals - Cosine Reduction Formula
17Calculus
This page covers the derivation and use of the cosine reduction formula for integration.
Recommended Books on Amazon (affiliate links) | ||
|---|---|---|
 |
 |
 |
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
The Organic Chemistry Tutor - 2578 video solution
video by The Organic Chemistry Tutor |
|---|
Log in to rate this practice problem and to see it's current rating. |
|---|
\(\int{ \cos^3 x ~dx }\)
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
video by The Organic Chemistry Tutor |
|---|
Log in to rate this practice problem and to see it's current rating. |
|---|
trig integration 17calculus youtube playlist
Really UNDERSTAND Calculus
Log in to rate this page and to see it's current rating.
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 |
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 | ||
Related Topics and Links
related topics on other pages |
|---|
external links you may find helpful |
To bookmark this page and practice problems, log in to your account or set up a free account.
Search Practice Problems
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.
|
| |
free ideas to save on bags & supplies |
|---|
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 |
|
|---|
