Trigonometric substitution example

Assuming , evaluate .

Solution

We start by completing the square:

Now we have

We make the substitution , where or , to get

Since , this reduces to

In order simplify the absolute value, we must look back at our substitution and our initial assumption that . In fact,

It follows that and the absolute value bars may be removed.

 

Continuing...

Finally, we can integrate...

 

To convert back to the variable , we use the right triangle where the angle has adjacent side , hypotenuse , and opposite side . It follows that