integration by parts:
how do u know the correct "u" and "dv" substitutions
Stacey Warren - Expert brainly.com
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While this is probably not the best answer in some ways, you could say that the correct ones are the ones that work. In other words, you are really using the integration by parts formula to write your given integral in terms of another integral. You chose the correct u and dv if the second integral is one that can be done.
The problem is that there is no real "rule" that says : "This is the choice to make". As I often tell people, no matter what pattern you think you see I can probably find an example that will break the pattern. However, having said that for many of the problems in a standard Calculus course the integrand tends to be a polynomial times an exponential, sine or cosine. In most of those cases (and note that I did say most!) the u is the polynomial and the dv is the exponential/sine/cosine, but again I can find examples where that may not quite work out in that manner.
One thing to try when making the choice is to simply ask yourself : "Is this choice liable to given an easier second integral?" If yes, try it, if not maybe try somethign else. For example with a polynomial times an exponential/sine/cosine the problem is the polynomial and if you let that be u then after differentiating the degreee of the polynomial in du will go down by one and so the second integral should be easier to do. Also recall that often you have to do integration by parts again on the second integral.
As I said, I realize that is not the best answer. It still doesn't really help with making the choices. Once you do enough examples you will get better at making the choices.
There are a few example at
that may help a little with seeing the "best" choices.