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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 = 2x + 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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