Trigonometric Substitutions

A variation on the substitution technique

Recall the antiderivatives of some trig functions

To solve integrals involving , , and , we need trigonometric substitutions

Definition

Expression Substitution Relevant identity

Example

Evaluate

solution
Since , we let . Differentiating this gives . Rearranging gives

(see Pythagorean identities)

Since , is always positive, which gives:

(see integration rules)

Now we have to return to monke . We can either draw a triangle (see next example), or observe that

Continuing,

where we used the fact that (see log laws), and set .

Example

Evaluate

solution
Let

Now we use substitution

Let

Since we have , that means . We can draw this on a triangle, and use Pythagorean theorem to calculate the bottom edge.

This means we can write in terms of :