How to Solve Integrals Using Integration by Parts

By Mark Ryan

You can think of integrating by parts as the integration version of the product rule for differentiation. The basic idea of integration by parts is to transform an integral you cant do into a simple product minus an integral you can do. Here’s the formula:

Integration by parts formula:

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And here’s a memory aid for it: In the first two chunks,

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the u and v are in alphabetical order. If you remember that, you can remember that the integral on the right is just like the one on the left, except the u and v are reversed.

Don’t try to understand the formula yet. You’ll see how it works in a minute. And don’t worry about understanding the first example until you get to the end of it. The integration by parts process may seem pretty convoluted your first time through it, so you’ve got to be patient. After you work through a couple examples, you’ll see it’s really not that bad at all.

The integration by parts box: The integration by parts formula contains four things: u, v, du, and dv. To help keep everything straight, organize your problems with a box like the one shown here.

The integration by parts box.
The integration by parts box.

For the first example, try

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The integration by parts formula will convert this integral, which you can’t do directly, into a simple product minus an integral you’ll know how to do. First, you’ve got to split up the integrand into two chunks — one chunk becomes the u and the other the dv that you see on the left side of the formula. For this problem, the ln (x) will become your u chunk. Then everything else is the dv chunk, namely

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After rewriting the above integrand, you’ve got the following for the left side of the formula:

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Now it’s time to do the box thing. For each new problem, you should draw an empty four-square box, then put your u (ln (x) in this problem) in the upper-left square and your

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in the lower-right square, as shown in the next figure.

Filling in the box.
Filling in the box.

Next, you differentiate u to get your du, and you integrate dv to get your v. The arrows in the second figure remind you to differentiate on the left and to integrate on the right. Think of differentiation — the easier thing — as going down (like going downhill), and integration — the harder thing — as going up (like going uphill).

Now complete the box:

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The completed box for

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is shown in the next figure.

The completed box.
The completed box.

You can also use the four-square box to help you remember the right side of the integration-by-parts formula: start in the upper-left square and draw (or just picture) a number 7 going straight across to the right, then down diagonally to the left, as shown in the next figure.

A box with a 7 in it. Who says calculus is rocket science?
A box with a 7 in it. Who says calculus is rocket science?

Remembering how you “draw” the 7, look back to the previous figure. The right side of the integration-by-parts formula tells you to do the top part of the 7, namely

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minus the integral of the diagonal part of the 7,

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By the way, all of this is much easier to do than to explain. Try it. You’ll see how this four-square-box scheme helps you learn the formula and organize these problems.

Ready to finish? Plug everything into the formula:

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In the last step, you replace the  

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times any old number is still just any old number.