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`

Graph of y(x)=sqrt(x^2-16)/x^2 - question involves integration by trigonometric substitution

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))`