Trade One Integral for an Easier One

You'll learnwhere the parts formula comes from, how to choose u and dv so the trade is worth making, and how to carry the minus sign through to the end.

Substitution handles products where one factor is the derivative of the other's inside. Integration by parts handles the rest, and it evaluates nothing: it trades your integral for a different one. So it never succeeds or fails on its own — it succeeds when what you are left with is easier than what you started with, and the same formula hands back something worse the moment u and dv are chosen the other way round. Twelve short steps; press Next for each one.

12 STEPS · 6 QUICK CHECKS · NOTHING IS EVALUATED, ONE INTEGRAL BECOMES ANOTHER · v2

trade one integral for another, and only a simpler one counts

Before you start

What you're looking at

One plot panel. Blue is always the integral you started with; violet is the one parts leaves you with; pink is an antiderivative and yellow the measured slope of a candidate answer.

The question

The formula swaps one integral for another and evaluates nothing. How do you tell whether the swap was worth making?

Step 1 / 12