- 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

Don't forget the first method we learned to find integrals: "thinking backwards." Sometimes you don't need substitution, parts, or partial fractions - you can simplify the integral and immediately see what to do with it.

We don't need anything fancy to find

Simplify the integral by squaring the integrand and then separating it out:

Then integrate each term:

Depending on how comfortable you are with thinking backwards, you might be able to do this one in your head:

However, you're still doing substitution behind the scenes, letting *u* = 2*x* + 3.

There's no reasonable way to think backwards from

That's what we learned integration by parts for!