# Thinking Backwards

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.

### Sample Problem

We don't need anything fancy to find

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

Then integrate each term:

### Sample Problem

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.

### Sample Problem

There's no reasonable way to think backwards from

That's what we learned integration by parts for.

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