In differential calculus, we derived formulas to differentiate products of functions.
Integration by Parts is a method to do the same sort of thing, for integrals. The formula can be derived simply by re-arranging the product rule (shown above).
In order to see this a bit better, we'll perform the following substitution:
The Fundamental Theorem of Calculus tells us that
Let's substitute! Replacing all occurrences of we get:
Now, there's a big ugly here - let's get rid of it! Remember - integration kills differentiation, and vice-versa!
So, we integrate both sides.
Rearranging to isolate , we see that
And, this is integration by parts!