Trigonometric substitution is used to replace radicals in integrals integrals with trigonometric functions, specifically replacing expressions of the forms , , and .

When seeing quadratics in a square root always try to complete the square to see if a trigonometric substitution is possible

Case 1 - , where

Substitute and .

is limited to the domain of and thus the domain of is

Because on the interval , we can just remove the absolute bars.

Examples

Example 1

Compute , where

Example 2

Compute

Case 2 - or , where

Substitute and .

is limited to the domain of and thus the domain of is

Because and is positive in the interval we can remove the absolute bars.

Example

Compute

Case 3 - , where

Substitute and .

is limited to the domain of and thus the domain of is

Since is positive on the interval we can again remove the absolute bars.

Example

Compute