Page 199 - Calculus Workbook For Dummies

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Chapter 10: Integration: Reverse Differentiation

Finding Antiderivatives: The

Guess and Check Method

Your textbook, as well as the Cheat Sheet in Calculus For Dummies, lists a set of
1 1
antiderivatives that you should memorize, such as sinx, , or 1 + x 2 . (Most of them
x
are simply the basic derivative rules you know written in reverse.) When you face
a problem that’s similar to one of these — like finding the antiderivative of sin x5
1
or — you can use the guess and check method: just guess your answer, check it by
8 x
differentiating, then if it’s wrong, tweak it till it works.

Q. What’s # sin x dx? taking its derivative with the chain rule,
3
you get sin x3 3 , which is what you want
A. - 1 cos x except for that first 3. To compensate
3
3 for that, simply divide your guess by 3:
3
- cos x
You’ve memorized that cosx- is the anti- 3 . That’s it. If you have any doubts
derivative of sinx — because, of course, about this second guess, take its derivate
-
sinx is the derivative of cosx. So a good and you’ll see that it gives you the desired
guess for this antiderivative would be integrand, sin x3 .
- cos x. When you check that guess by
3

3 14. What’s # 2
6
13. Determine #^ 4 x - 1h dx. sec x dx?
Solve It Solve It