8. Integration by Trigonometric Substitution

by M. Bourne

In this section, we see how to integrate expressions like


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


Example 2


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


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))`