Nothing New, and That Is the Difficulty

You'll learnto run four questions in order over an integrand you have never seen, to read the structural tell that decides between rewriting, substitution and parts, to tell what a failed attempt actually condemns — and to settle every one of those answers in seconds by differentiating what you wrote and comparing it with the integrand.

Every technique in this unit is now available and no new one is coming. What is new is that nobody tells you which one to use, and the integrands that call for different methods can differ by a single symbol. So the skill stops being a procedure and becomes a reading: what is inside what, whose derivative is present, which two pieces are unlike each other. And whichever road you take, one test settles it — differentiate the answer and see whether the integrand comes back.

8 STEPS · 6 QUICK CHECKS · READ THE STRUCTURE, THEN DIFFERENTIATE THE ANSWER · v1

Read the structure, choose the technique, then differentiate the answer

Before you start

What you're looking at

One plot panel and one board. The blue curve is always the integrand — the thing being antidifferentiated. The board beside it writes that integrand out and lights the part of it that decides the technique. A yellow dashed curve is the measured slope of a candidate answer; a red one is the measured slope of a wrong answer.

The question

With every technique already available and no new one coming, how do you tell from the integrand alone which one to reach for — and what do you do when the first choice does not work?

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 choice costs, drawn to scale.

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