Integration by substitution is a way of undoing the chain rule. This is a once-in-a-lifetime opportunity to learn derivatives inside and out, forwards and backwards. Exciting, eh? Learning integration by substitution is almost as exciting as winning Deal or No Deal. Given a derivative that was produced by the chain rule, integration by substitution lets us work backwards to find an antiderivative. This method is also called u-substitution because we usually use u instead of some other letter.
We've done some integration by substitution already. We call it thinking backwards.
We can do integration by substitution using either Lagrange notation (primes u' ) or Leibniz notation (fractions like 
). While your teacher might have a preference, we don't care which of these ways you use. Just be careful not to mix them up. You can use primes (u' ) or you can use fractions 
, but don't use both in the same problem.