Do the Algebra First

You'll learnwhich rewriting a rational integrand is asking for — long division, a completed square, or a substitution — and how to settle each one by differentiating what you wrote.

Some integrands have no antiderivative you recognize until they are rewritten, and after the rewrite they are routine. The whole difficulty is the algebra step, which is also the step that gets skipped, because the fraction looks like something the power rule ought to handle. Nothing here is a new antiderivative; it is a question of which rewriting carries you back to a rule you already have. Twelve short steps; press Next for each one.

12 STEPS · 6 QUICK CHECKS · PREPARE THE INTEGRAND, THEN DIFFERENTIATE WHAT YOU WROTE · v2

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

Before you start

What you're looking at

One plot panel. Blue is always the integrand; pink is what the algebra turns it into; yellow is the measured slope of a candidate answer and red the slope of one that skipped a step.

The question

Which rewriting does a rational integrand need before it can be integrated at all, and how do you check what you wrote?

Step 1 / 12