Example 6.4.2. Using Trigonometric Substitution.
Evaluate
Solution 1.
We begin by noting that and hence If we let then
Setting gives We are almost ready to substitute. We also wish to change our bounds of integration. The bound corresponds to (for when ). Likewise, the bound of is replaced by the bound Thus
On is always positive, so we can drop the absolute value bars, then employ a power-reducing formula:
This matches our answer from before.