How to Do Integration by Parts More than Once
Sometimes you have to use the integration-by-parts method more than once because the first run through the method takes you only part way to the answer.
Go down the LIATE list and pick your u.
Using the mnemonic device LIATE:
To pick your u, go down this list in order; the first type of function on this list that appears in the integrand is the u. Then everything else is automatically the dv.
Organize the problem using the first box shown in the figure below.
In the second box above, your u and dv from Step 1 have been put in the u and dv cells. The third box shows the completed breakdown of the problem after differentiating the u and integrating the dv.
Use the integration-by-parts formula.
You end up with another integral,
that can’t be done by any of the simple methods — reverse rules, guess and check, and substitution. But note that the power of x has been reduced by one, so you’ve made some progress.
Integrate by parts again.
(If you feel like it, you can change the negative C into a positive C because C is just any old constant.)
Take the result from Step 4 and substitute it for the
in the answer from Step 3 to produce the whole enchilada.
(In this last line, you can replace the 2C with C because twice any old constant is the same thing as any old constant. This looks weird mathematically, but, don’t worry, it’s okay.)