The chain rule, run backwards

You'll learnto spot the composition whose inside brings its own derivative, to rewrite an integral so that no trace of the old variable survives, to supply a missing constant and to recognise that a missing variable cannot be supplied — and to finish a definite integral by one of exactly two routes rather than by a mixture of both.

Differentiating a composition multiplies by the derivative of the inside. Substitution looks for that shape in an integrand and runs the machine in reverse: name the inside u, spend the inner derivative as du, and the integral becomes something you already know. Two things go wrong, and they are independent. The differential does not match — a stray x, a dropped constant. Or the limits never get converted, and that one is silent: it returns an ordinary-looking number with nothing visibly wrong about it.

8 STEPS · 6 QUICK CHECKS · CHANGE THE AXIS, KEEP THE AREA · v1

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 drawn on one number line. The upper one reads that line as x and carries the integrand you were handed. The lower one reads the same line as u and carries the integral after the change of variable. Both are the same rectangle at the same scale, so a region in one can be compared with a region in the other by eye.

The question

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

Watch for

Every number on screen is integrated from the same functions the curves are drawn from, so the two areas are free to disagree. Where a wrong answer is shown it is drawn as a region, in red, on the same axis as the right one.

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