Mastering Integration by Parts

Integration by parts is actually just the product rule running in reverse. When you are stuck

with a messy integral of two functions multiplied together, like x times sine x,

you cannot just guess. You have to split them up. Call one piece u

and the other dv, then swap them using the formula uv minus the integral of v du.

It turns a nightmare problem into a simple subtraction game. Even for those nasty trig powers,

stop brute forcing it. Use identities to break products into sums.

The secret to calculus isn't being smart, it's being lazy enough to rewrite the problem.