- Topics At a Glance
- Indefinite Integrals Introduction
**Integration by Substitution: Indefinite Integrals**- Legrange (Prime) Notation
- Leibniz (Fraction) Notation
- Integration by Substitution: Definite Integrals
- Integration by Parts: Indefinite Integrals
- Some Tricks
- Integration by Parts: Definite Integrals
- Integration by Partial Fractions
- Integrating Definite Integrals
- Choosing an Integration Method
- Integration by Substitution
- Integration by Parts
- Integration by Partial Fractions
- Thinking Backwards
- Improper Integrals
- Badly Behaved Limits
- Badly Behaved Functions
- Badly Behaved Everything
- Comparing Improper Integrals
- The
*p*-Test - Finite and Infinite Areas
- Comparison with Formulas
- In the Real World
- I Like Abstract Stuff; Why Should I Care?
- How to Solve a Math Problem

**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

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.