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You CAN Ace Calculus

17calculus > integrals > trig integration > sine-cosine integration

Topics You Need To Understand For This Page

Trig Identities and Formulae - NEW

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 - NEW

basic trig derivatives

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

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

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

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

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

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

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 (where n is a positive integer)

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

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

Reduction Formulas (where n is an integer and \(n>1\))

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

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

links

17calculus trig formulas - full list

Calculus Main Topics

Tools

Related Topics and Links

Sine-Cosine Trig Integration

on this page: ► one angle     ► reduction formulas     ► two angles

Difference Between Trig Integration and Trig Substitution

Trig integration, covered on this page, is the evaluation of integrals that already have trig functions in the integrand.

Trig substitution is a technique that takes an integrand that most likely does NOT contain any trig functions, and uses some trig identities to introduce trig functions into the integrand. Once the integral is completely transformed, then trig integration is used to evaluate the integral. Once the evaluation is complete, another set of substitutions, based on the original ones, is done to convert the result back to the original variable.

This page covers integration of functions involving sines and/or cosines in more advanced form that require techniques other than just integration by substitution. [If you are first learning sine and cosine in integration, check out the basics of trig integration page.]

Trig Integration - Complete Case List Summary (One Angle)

Trig Integration - Complete Case List Summary (One Angle)

\(\int{\sin^mx \cos^nx~dx}\)

\(n=0\)

\(\int{\sin^mx~dx}\)

use reduction formula

\(m=0\)

\(\int{\cos^mx~dx}\)

use reduction formula

odd m

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

factor out \(\sin x\), use \(\sin^2x=1-\cos^2x\) and let \(u=\cos x\)

odd n

\(\int{\sin^mx\cos^{2k+1}x~dx}\)

factor out \(\cos x\), use \(\cos^2x=1-\sin^2x\) and let \(u=\sin x\)

even m and n

\(\int{\sin^{2k}x\cos^{2p}x~dx}\)

use half-angle formulas

\(\int{\sec^mx \tan^nx~dx}\)

\(n=0\)

\(\int{\sec^mx~dx}\)

use reduction formula

\(m=0\)

\(\int{\tan^nx~dx}\)

use reduction formula or use \(\sec^2x=1+\tan^2x\), expand out
and try one of the following two cases

even m

\(\int{\sec^{2k}x\tan^nx~dx}\)

factor out \(\sec^2x\), use \(\sec^2x=1+\tan^2x\) and let \(u=\tan x\)

odd n

\(\int{\sec^mx\tan^{2k+1}x~dx}\)

factor out \(\sec x\tan x\), use \(\sec^2x=1+\tan^2x\) and let \(u=\sec x\)

none of the above 4 cases hold

convert trig functions to sine and cosine and
try the sine/cosine techniques

In order to choose the technique you need to use, you need to determine the form of the integrand and how many angles are involved.

One Angle

When all the sine and cosine terms in the integrand involve the same angle, here is what you do.

Case 1 ( one sine term only ): \( \int{ \sin^n(\theta) ~d\theta} \)

Use the reduction formula found in the section below.

 

Case 2 ( one cosine term only ): \( \int{ \cos^n(\theta) ~d\theta} \)

Using the same technique found in the reduction section below,
you can get a reduction formula for cosine.

 

Case 3: \( \int{ \sin^m(\theta)~\cos^n(\theta) ~d\theta} \) with odd m or n

Factor out the odd term (if they are both odd, you can choose) and
use \(\sin^2\theta + \cos^2\theta = 1\) on the remaining term,
then use substitution.

 

Case 4: \( \int{ \sin^m(\theta)~\cos^n(\theta) ~d\theta} \) with even m and n

Use the half-angle formulas to remove the powers.

Reduction Formulas

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 \(\sin^2 x\). The formula is derived on a separate page. The reduction formula for cosine is derived on this page.

Reduction Formulas (where n is a positive integer)

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

derivation

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

derivation

Two Angles

When you have different angles, in addition to the double-angle formulas at the top of the page, use the following formulas to simplify the integrand.

\(\displaystyle{ \sin(\alpha)\sin(\theta) = \frac{1}{2}\left[ \cos(\alpha - \theta) - \cos(\alpha + \theta) \right] }\)

\(\displaystyle{ \cos(\alpha)\cos(\theta) = \frac{1}{2}\left[ \cos(\alpha - \theta) + \cos(\alpha + \theta) \right] }\)

\(\displaystyle{ \sin(\alpha)\cos(\theta) = \frac{1}{2}\left[ \sin(\alpha + \theta) + \sin(\alpha - \theta) \right] }\)

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Practice Problems

Instructions - - Unless otherwise instructed, evaluate the following integrals using the techniques on this page. Give all answers in exact, simplified form.

Here are a few practice problems. You can find more practice problems on this page.

Level A - Basic

Practice A01

\(\displaystyle{\int{\sin^4(2x)\cos(2x)~dx}}\)

answer

solution

Practice A02

\(\displaystyle{\int{\sin(4x)\cos(2x)~dx}}\)

solution

Practice A03

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

solution

Practice A04

\(\displaystyle{\int{\cos^4x\sin^3x~dx}}\)

solution


Level B - Intermediate

Practice B01

\(\displaystyle{\int{\cos^4x~\sin^2x~dx}}\)

answer

solution

Practice B02

\(\displaystyle{\int_{0}^{\pi/2}{\sin^7\theta~\cos^5\theta~d\theta}}\)

solution

Practice B03

\(\displaystyle{\int{\cos^4x~dx}}\)

solution


Level C - Advanced

Practice C01

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

solution

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