Math basics › Calculus II
Integration by parts
The formula
∫ u dv = uv − ∫ v du. It is the product rule run backwards, for an integral of two different kinds of function multiplied together.
Choosing u
Pick as u the part that gets simpler when you differentiate it. LIATE is the usual guide: logs, inverse trig, algebraic (powers of x), trig, exponentials. Earlier in that list makes a better u.
Everything else is dv
dv includes the dx. You integrate dv to get v, so it has to be something you can integrate.
When it goes in a circle
With eˣ sin x, two rounds bring the original integral back. Move it to the other side and solve for it like an unknown.