In differential calculus, we derived formulas to differentiate products of functions.

Recall that

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 $\textcolor{teal}{f(x)},$ we get:

Now, there's a big ugly $\frac{d}{dx}$ here - let's get rid of it! Remember - integration kills differentiation, and vice-versa!

So, we integrate both sides.

Rearranging to isolate $\int \textcolor{orchid}{h(x)}~\textcolor{green}{g\left(x\right)}\ dx$, we see that

And, this is integration by parts!