# 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.**dv*.Organize the problem using the first box shown in the figure below.

In the second box above, your

*u*and*dv**u*and*dv**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 2*C* 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.)