How to Do Integration by Parts, Integrating by parts is the integration version of the product rule for differentiation. The basic idea of integration by parts is to transform an integral you [i]can’t[/i] do into a simple product minus an integral you [i]can[/i] do. Here’s the formula:[br][br][img width=264,height=29]https://www.dummies.com/wp-content/uploads/202716.image0.png[/img][br][br]Don’t try to understand this yet. Wait for the examples that follow:[br][img width=384,height=29]https://www.dummies.com/wp-content/uploads/202717.image1.png[/img][br]If you remember that, you can easily remember that the integral on the right is just like the one on the left, except with the [i]u[/i] and [i]v[/i] reversed. Here’s the method in a nutshell.[br][br][img width=139,height=31]https://www.dummies.com/wp-content/uploads/202718.image2.png[/img][br][br]First, you’ve got to split up the integrand into a [i]u[/i] and a [i]dv[/i] so that it fits the formula. For this problem, choose ln([i]x[/i]) to be your [i]u. [/i]Then, everything else is the [i]dv,[/i] namely[img width=41,height=24]https://www.dummies.com/wp-content/uploads/202719.image3.png[/img]Next, you differentiate [i]u[/i] to get your [i]du[/i], and you integrate [i]dv[/i] to get your [i]v[/i]. Finally, you plug everything into the formula and you’re home free.[br][br][br][img width=259,height=259]https://www.dummies.com/wp-content/uploads/202720.image4.jpg[/img][br][br][br]The integration by parts box. To help keep everything straight, organize integration-by-parts problems with a box like the one in the above figure. Draw an empty 2-by-2 box, then put your [i]u[/i], ln([i]x[/i]), in the upper-left corner and your [i]dv[/i],[img width=41,height=25]https://www.dummies.com/wp-content/uploads/202721.image5.png[/img]in the lower-right corner, as in the following figure.[br][br][img width=421,height=259]https://www.dummies.com/wp-content/uploads/202722.image6.jpg[/img][br][br]Filling in the box. The arrows in this 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).[br][br]Now complete the box:[br][br][img width=188,height=225]https://www.dummies.com/wp-content/uploads/202723.image7.png[/img][br][br]The completed box for[img width=80,height=31]https://www.dummies.com/wp-content/uploads/202724.image8.png[/img]is shown in the next figure.[br][br][br][img width=421,height=259]https://www.dummies.com/wp-content/uploads/202725.image9.jpg[/img][br][br]A good way to remember the integration-by-parts formula is to start at the upper-left square and draw an imaginary number 7 — across, then down to the left, as shown in the following figure. This is an oh-so-evenly mnemonic device.[br][br][br][img width=259,height=259]https://www.dummies.com/wp-content/uploads/202726.image10.jpg[/img][br][br]Remembering how you draw the 7, look back to the figure with the completed box. The integration-by-parts formula tells you to do the top part of the 7, namely[img width=71,height=48]https://www.dummies.com/wp-content/uploads/202727.image11.png[/img]minus the integral of the diagonal part of the 7,[img width=80,height=48]https://www.dummies.com/wp-content/uploads/202728.image12.png[/img][br][br]Try the box technique with the 7 mnemonic. You’ll see how this scheme helps you learn the formula and organize these [br][br]problems.)Ready to finish? Plug everything into the formula:[br][br][br][img width=308,height=321]https://www.dummies.com/wp-content/uploads/202729.image13.png[/img]