Math 1B  Homework Solution 5.6#22

This integral can be tackled in a number of different ways.  In class, we did it by first by trigonometric substitution and then by parts.

 

Method 1: Trigonometric Substitution

 

The idea is to simplify the radical by using Pythagorus’ theorem to express the sum of squares as a single square, thus eliminating the radical.  In the case with a sum of squares, use the  form of the identity, or, in this case, .  So we want .  With that substitution,  and .  Putting these together,

 

 

 

This may not seem much improved, or even worse, from where we started, but it certainly is different!  Observe, nonetheless, how we can rewrite this integral now using the Pythagorean identity:

 

 

The second integral has a simple antiderivative and the first has a fairly obvious substitution: . 

 

Thus  

 

Method 2:  Integration by parts. 

 

This method requires some perception and/or guess/check to get the right substitution:

 

Whence  yields