Differential Equations For Dummies
Book image
Explore Book Buy On Amazon

The guess-and-check method works when the integrand — that’s the thing you want to antidifferentiate (the expression after the integral symbol, not counting the dx) — is close to a function that you know the reverse rule for. For example, say you want the antiderivative of cos(2x). Well, you know that the derivative of sine is cosine. Reversing that tells you that the antiderivative of cosine is sine. So you might think that the antiderivative of cos(2x) is sin(2x). That’s your guess. Now check it by differentiating it to see if you get the original function, cos(2x):

image0.png

This result is very close to the original function, except for that extra coefficient of 2. In other words, the answer is 2 times as much as what you want. Because you want a result that’s half of this, just try an antiderivative that’s half of your first guess:

image1.png

Check this second guess by differentiating it, and you get the desired result. Thus, the antiderivative of cos(2x) is

image2.png
  1. Guess the antiderivative.

    This looks sort of like a power rule problem, so try the reverse power rule.

    image3.png
  2. Check your guess by differentiating it.

    image4.png
  3. Tweak your first guess.

    image5.png
  4. Check your second guess by differentiating it.

    image6.png

This answer checks — you’re done! The antiderivative of

image7.png

The guess-and-check method works well when the function you want to antidifferentiate has an argument like 3x or 3x + 2 (where x is raised to the first power) instead of a plain old x.

image8.png

You just have to tweak your guess by the reciprocal of the coefficient of x — the 3 in 3x + 2, for example (the 2 in 3x + 2 has no effect on your answer). In fact, for these easy problems, you don’t really have to do any guessing and checking. You can immediately see how to tweak your guess. It becomes sort of a one-step process.

About This Article

This article is from the book:

About the book author:

Steven Holzner was an award-winning author of more than 130 books, of which more than 2 million copies have been sold. His books have been translated into 23 languages. He served on the Physics faculty at Cornell University for more than a decade, teaching both Physics 101 and Physics 102. Holzner received his doctorate in physics from Cornell and performed his undergraduate work at Massachusetts Institute of Technology, where he also served as a faculty member.

This article can be found in the category: