Trade one integral for an easier one

You'll learnto derive the parts formula by moving one term of the product rule across an equals sign, to choose u and dv by what differentiating and integrating each one actually does, to carry the minus sign and the bracket through a repeated application, and to evaluate the boundary term of a definite integral in place.

Substitution handles products where one factor is the derivative of the other's inside. Integration by parts handles the rest, and it does not evaluate anything: it trades your integral for a different one. So the method never succeeds or fails on its own — it succeeds when the integral you are left with is easier than the one you started with, and the same formula, applied correctly, hands back something worse the moment u and dv are chosen the other way round. The whole skill is making the trade profitable, and the whole failure is making it worse.

8 STEPS · 6 QUICK CHECKS · NOTHING IS EVALUATED, ONE INTEGRAL BECOMES ANOTHER · v1

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

Before you start

What you're looking at

One plot panel. The blue curve is always the integral you STARTED with — the thing under the integral sign. The violet curve or band is the integral parts leaves you WITH. Pink is an antiderivative. A yellow dashed curve is the measured slope of a candidate answer, and a red one is the measured slope of a wrong one.

The question

The formula swaps one integral for another and evaluates nothing. So how do you tell, before doing any more work, whether the swap was worth making?

Watch for

Every slope on screen is a central difference of the candidate itself, taken at draw time, and every area is a Simpson sum of the function being drawn. A yellow curve lying on the blue one is a claim this picture could have failed.

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