Trigonometric Substitution

Calculus 2

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Overview

Some integrals solve much more easily when we perform a trigonometric substitution. In particular, trigonometric substitution is great for getting rid of radicals in our integrand (given that our tried and true u-substitution isn't an option).

There are three substitutions we make use of:

$x = \frac{a}{b}\sin(\theta) \ \ \ \ x=\frac{a}{b}\sec(\theta) \ \ \ \ x=\frac{a}{b}\tan(\theta)$

The goal when making one of these substitutions is to obtain a Pythagorean identity for which the radical will reduce (although the radical is not necessary, as we will see later):

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Things to keep in mind:

We are making a substitution for $x$ , so also replace $dx$
Always put our answer back into terms of $x$
When simplifying, try rewriting everything in terms of sines and cosines
Don't forget to consider the angle restrictions (more on this when we discuss definite integrals)

Indefinite Integrals

Example 1 Example 2 Example 3 Exmaple 4

Note that trigonometric substitution can also be used in cases where there isn’t a radical in the integrand.

example 5

Challenge 1 Challenge 2

Example 1

Compute $\int \sin^3x \cos x \ dx$ .

View Answer

$\frac{1}{4}\sin^4x +C$ or $\frac{1}{4} \cos^4x - \frac{1}{2}\cos^2x + C$

View Solution

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This video tutorial is still in the works. In the meantime, please use the text version.

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We Text

\begin{aligned} \int \sin^3x \cos x dx &= \int u^3 du \\ &= \frac{1}{4} u^4 +C\\ &= \frac{1}{4} \sin^4x +C \end{aligned}

Definite Integrals

Practice Problems

Trigonometric Integrals

Numerical Integration