# 8. Integration by Trigonometric Substitution

by M. Bourne

In this section, we see how to integrate expressions like

int(dx)/((x^2+9)^(3//2))

Depending on the function we need to integrate, we substitute one of the following to simplify the expressions to be integrated:

For sqrt(a^2-x^2), use  x =a\ sin\ theta

For sqrt(a^2+x^2), use  x=a\ tan\ theta

For sqrt(x^2-a^2), use x=a\ sec\ theta

After we use these substitutions we'll get an integral that is "do-able".

### Example 1

int(dx)/((x^2+9)^(3//2))

### Example 2

int_4^5(sqrt(x^2-16))/(x^2)dx

### Exercises

Integrate each of the given functions:

1. intsqrt(16-x^2)dx

2. int(3\ dx)/(xsqrt(4-x^2))

3. int(dx)/(sqrt(x^2+2x))

Didn't find what you are looking for on this page? Try search:

### Online Algebra Solver

This algebra solver can solve a wide range of math problems. (Please be patient while it loads.)

Play a math game.

(Well, not really a math game, but each game was made using math...)

Sign up for the free IntMath Newsletter. Get math study tips, information, news and updates each fortnight. Join thousands of satisfied students, teachers and parents!

Given name: * required

Family name:

email: * required

See the Interactive Mathematics spam guarantee.