The Chain Rule, Run Backwards

You'll learnto spot the composition whose inside brings its own derivative, to rewrite an integral so no x survives, and to finish a definite one by exactly two routes.

Differentiating a composition multiplies by the derivative of the inside. Substitution finds that shape in an integrand and runs the machine backwards: name the inside u, spend the inner derivative as du, and what is left is something you already know. Two things go wrong. The differential does not match, or the limits never get converted — and the second one is silent. Twelve short steps; press Next for each one.

12 STEPS · 6 QUICK CHECKS · CHANGE THE AXIS, KEEP THE AREA · v2

set u to the inside, spend the inner derivative as du, push the limits through

Before you start

What you're looking at

Two panels on one number line. The upper reads it as x and carries the integral you were handed; the lower reads it as u and carries the same integral after the change of variable.

The question

The two regions have different shapes. Do they have the same area, and what has to be true for that to hold?

Step 1 / 12