How to Find Antiderivatives Using Reverse Rules

By Mark Ryan

You can use reverse rules to find antiderivatives. The easiest antiderivative rules are the ones that are the reverse of derivative rules you already know. These are automatic, one-step antiderivatives with the exception of the reverse power rule, which is only slightly harder.

You know that the derivative of sin x is cos x, so reversing that tells you that an antiderivative of cos x is sin x. What could be simpler? But don’t forget that all functions of the form sin x + C are antiderivatives of cos x. In symbols, you write

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This table lists the reverse rules for antiderivatives.

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You can also use the (slightly more difficult) reverse power rule. By the power rule for differentiation, you know that

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Here’s the simple method for reversing the power rule. Use y – 5x4 for your function. Recall that the power rule says to

  1. Bring the power in front where it will multiply the rest of the derivative.

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  2. Reduce the power by one and simplify.

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To reverse this process, you reverse the order of the two steps and reverse the math within each step. Here’s how that works for the above problem:

  1. Increase the power by one.

    The 3 becomes a 4.

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  2. Divide by the new power and simplify.

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Of course, all the best rules contain some kind of caveat, and this is no exception.

The reverse power rule does not work for a power of negative one. The reverse power rule works for all powers (including negative and decimal powers) except for a power of negative one. Instead of using the reverse power rule, you should just memorize that the antiderivative of

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(rule 3 in the table).

Test your antiderivatives by differentiating them. Especially when you’re new to antidifferentiation, it’s a good idea to test your antiderivatives by differentiating them — you can ignore the C. If you get back to your original function, you know your antiderivative is correct.

With the antiderivative you just found and using the fundamental theorem, you can determine the area under 20x3 between, say, 1 and 2:

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