Do the Algebra First

You'll learnto compare degrees before touching a rational integrand, to divide an improper fraction into a polynomial plus a leftover, to complete the square and read the arctangent form off it with the right coefficient and the right shift, to keep a numerator that splits apart from a denominator that never does — and to settle every one of those in seconds by differentiating what you wrote.

Some integrands have no antiderivative you recognise until they are rewritten, and after the rewrite they are routine. The difficulty is entirely in the algebra step, which is also the step that gets skipped, because the integrand looks like something the power rule ought to handle. Nothing in this topic is a new antiderivative. It is all a question of which rewriting carries you back to a rule you already have — and of noticing, before you start, which rewriting the integrand is asking for.

8 STEPS · 6 QUICK CHECKS · PREPARE THE INTEGRAND, THEN DIFFERENTIATE WHAT YOU WROTE · v1

Compare the degrees, complete the square, then differentiate what you wrote

Before you start

What you're looking at

One plot panel. The blue curve is always the integrand — the thing being integrated. Pink is what the algebra turns it into: a quotient, a completed square, a valid rewriting. The yellow dashed curve is the measured slope of a candidate answer, and the red one is the measured slope of an answer that skipped a preparation.

The question

Which rewriting does a rational integrand need before it can be integrated at all, and how do you know in seconds whether the thing you wrote down is right?

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 skipped preparation costs.

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