[MrRubix]

Integration by parts is actually really easy. All you really need to know is the whole, like

integral of udv = uv - integral of vdu (just to clarify, udv means u times the derivative of v)

Basically you use it when you're being asked to find the integral of (usually) the product of two functions that are (usually) of different forms such that u-sub and other types of integration don't seem appropriate. For example, if I asked you to find the integral of e^x*x^2, you might wanna use int. by parts.

In such cases, you usually set up the following four things: u, du, v, and dv. You generally want to set u to something with a simple derivative and you want to set dv to something that has an easy integral. So in this case, you might want to make u=x^2 and dv=e^x. Therefore du is 2x and v is e^x. We wouldn't really want to set u=e^x and dv=x^2 because du is still e^x, whereas v would become x^3/3, which isn't really any simpler than before. So choose your u and dv wisely.

integral of udv = uv - integral of vdu
integral of x^2*e^x dx = x^2*e^x - integral of e^x * 2x dx

The tricky part about int by parts questions is that sometimes the integral on the righthand side is, in itself, ANOTHER int by parts question. So looking at that integral separately, let's treat it as a new problem.

integral of e^x * 2x -> set u=2x and dv=e^x, so du=2 and v=e^x
integral of e^x * 2x dx = 2x*e^x - integral of e^x * 2 dx

We can always bring constants out of integrals:
integral of e^x * 2x dx = 2x*e^x - 2* integral of e^x dx, and we already know the integral of e^x, so:
integral of e^x * 2x dx = 2x*e^x - 2* e^x

So now we plug that back into our first equation:
integral of udv = integral of x^2*e^x dx = x^2*e^x - integral of e^x * 2x dx
integral of udv = integral of x^2*e^x dx = x^2*e^x - (2x*e^x - 2* e^x)
= e^x * (x^2-2x+2)

Anyways that's a basic example of int by parts. I typed that out of my ass so uh let me know if something doesn't make sense.