Question

What is the general method for integrating by parts?

Answer 1

`intu` `dv=uv-int` `v` `du`

Explanation:

The trick with doing integration by parts comes in choosing your `u` and `dv` accordingly. If you choose poorly, the problem will usually become harder.

For example, consider `intxsinx` `dx`.

If you let `u=sinx` and `dv=x` `dx` (and accordingly `du=cosx` `dx` and `v=1/2x^2`) you will end up with

`intxsinx` `dx=1/2x^2sinx-1/2intx^2cosx` `dx`.

You can try to evaluate the integral `intx^2cosx` `dx` but if you make similar `u` and `dv` choices this problem will continue to get more complicated.

Instead, for `intxsinx` `dx`, let `u=x` and `dv=sinx` `dx` (and accordingly `du=1` `dx` and `v=-cosx`). Then you get

`intxsinx` `dx=-xcosx-int-cosx*(1)` `dx`

`=-xcosx+intcosx` `dx=-xcosx+sinx+C`.

This way, you can actually do the problem.

Now you are probably thinking "Is there a way to know to to make `u` and `dv` so I don't have to go through this process of trial and error?"

The answer is yes. Use the acronym LIPET to remember the order of choosing `u`, in order of best to choose to worst to choose.

BEST
L: logarithmic stuff
I: inverse trig stuff
P: polynomialish stuff
E: exponential stuff
T: trig stuff
WORST

Trial and error should help explain why this order is helpful.