On-line Math 21

5.4  Techniques of integration

Example 4

ó õ xexdx

Solution

The game, and it is a game, is to pick the right parts to be u and dv . You have to be able to integrate whatever you choose as dv , and you have to hope that, when you differentiate u , it becomes simpler. Here, you couldn't ask for more. Set u = x , and then dv = e^xdx . I usually write down the other two pieces: du = dx and v = e^x . Don't bother with the +C here when you integrate dv , it won't really change anything.

Then, applying the equation above - which is, on one level, all there is to integration by parts - we see:

ó õ x exdx = ó õ u dv = u v- ó õ v du = x ex- ó õ exdx = x ex-ex+C.

Copyright (c) 2000 by David L. Johnson.