A Whole Family, and the One You Were Asked For

You'll learnto write an indefinite integral as the family it is, to run the power rule backwards including at the one exponent it refuses, to carry the reciprocal coefficient a linear inside demands — and to check every one of those in seconds by differentiating what you wrote and comparing it with the integrand.

A definite integral asks how much. An indefinite one asks which function, and the answer is never a single function: it is a whole family, spaced by constants, all with exactly the same derivative. The + C says so. Everything else in this topic is a derivative rule read from right to left, which means every answer can be checked by differentiating it back — and a wrong antiderivative is not wrong in the abstract, it is a curve whose slope comes out at the wrong height.

8 STEPS · 6 QUICK CHECKS · DIFFERENTIATE WHAT YOU WROTE AND COMPARE · v1

Antidifferentiate, keep the constant, then differentiate what you wrote

Before you start

What you're looking at

One plot panel. The blue curve is always the integrand — the function being antidifferentiated. The pink curves are antiderivatives of it. The yellow dashed curve is the measured slope of a candidate answer, and the red dashed one is the measured slope of a wrong answer.

The question

What does an indefinite integral actually name, and how do you know in seconds whether the thing you wrote down is one of them?

Watch for

Every slope on screen is a central difference of the candidate itself, taken at draw time. A yellow curve lying exactly on the blue one is a claim this picture could have failed; a red one standing away from it is what a wrong rule costs.

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