Summary

This is the method we use to find antiderivatives by using the chain rule in reverse. The book seems to make this rather complicated, and doesn't do a very good job of setting up the problem, I think. The idea is simply this:

If you want to rewrite this, to make it appear simpler, you could do so using a new variables u: if you set

then

or

Then we rewrite the integral as

When you have a perfect derivative in the integrand, it means that the integral is easy to do! It's easy to spot a function whose derivative is f'(u)!

#8, p. 401

There are some important special cases of this rule that our book points out, but I will write them differently from our text:

1. General Power rule:

   

   #10, p. 401

2. Exponentials:

   

   (You're looking at ``e of stuff time stuff prime'', which you recognize as a pure derivative.) #17, p. 401

3. Logarithmic derivatives:

   

   #33, p. 401

#39, p. 402