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.