# 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 trigonometric expressions to simplify the integration:

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".

Take note that we are not integrating trigonometric expressions (like we did earlier in Integration: The Basic Trigonometric Forms and Integrating Other Trigonometric Forms and Integrating Inverse Trigonometric Forms.

Rather, on this page, we substitute a sine, tangent or secant expression in order to make an integral possible.

### Example 1

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

### Example 2

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

The curve y=(sqrt(x^2-16))/(x^2), with the area under the curve between x=4 and x=5 shaded.

Continues below

## 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))

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