The Reverse Power Rule for Integration

What the Reverse Power Rule Is For

Differentiation and integration are closely connected. If the power rule tells you how to differentiate a power of , then the reverse power rule tells you how to work backward and find an antiderivative.

For example, differentiating gives . So when you see inside an integral, you can reverse that process to recover an antiderivative.

The key idea is simple: increase the exponent by 1, then divide by the new exponent. Finally, add the constant of integration.

The Formula

For any exponent other than :

This is called the reverse power rule because it reverses the ordinary power rule for differentiation.

Why the Rule Works

Start with the expression on the right side of the rule:

Now differentiate it using the ordinary power rule:

The factor cancels with the denominator , leaving exactly . That is why the reverse power rule works.

A Step-by-Step Method

Rewrite the integrand as a power of . This is especially important when you see roots or variables in denominators.
Increase the exponent by 1.
Divide by the new exponent.
Add the constant of integration, .
Differentiate your answer mentally to make sure you recover the original integrand.

Worked Examples

Example 1: Basic Power Rule

Find the indefinite integral:

Step 1: The exponent is .

Step 2: Increase the exponent by 1: .

Step 3: Divide by the new exponent:

Step 4: Add the constant of integration:

Result:

Example 2: A Constant Multiple

Find the indefinite integral:

Step 1: Keep the constant outside the integration.

Step 2: Increase the exponent from to .

Step 3: Divide by the new exponent:

Result:

Example 3: Fractional Exponents

Find the indefinite integral:

Step 1: Rewrite the square root as a fractional exponent:

Step 2: Add 1 to the exponent:

Step 3: Divide by the new exponent:

Result:

Example 4: Negative Exponents

Find the indefinite integral:

Step 1: Rewrite the denominator as a negative power:

Step 2: Add 1 to the exponent:

Step 3: Divide by the new exponent:

Result:

Example 5: A Polynomial

Find the indefinite integral:

Step 1: Integrate each term separately.

Step 2: Apply the reverse power rule to .

Step 3: Apply the reverse power rule to .

Step 4: The constant term integrates to .

Result:

The Important Exception: Exponent −1

There is one exponent for which the reverse power rule does not work: .

If you tried to apply the formula, you would have to divide by zero:

Instead, has its own special antiderivative:

More generally, when the integrand is , the antiderivative is , assuming the argument is nonzero.

Do not use the power rule for

The expression is the classic exception. Its antiderivative is , not something obtained by adding 1 to the exponent.

How to Recognize When You Need the Reverse Power Rule

On an AP Calculus problem, the reverse power rule may not appear in the convenient form . You should be comfortable rewriting expressions first.

Roots

Rewrite radicals as fractional powers:

Variables in the Denominator

Move powers from the denominator to the numerator:

Constants

Constant multiples can be pulled outside the integral:

Common Mistakes

Forgetting to divide by the new exponent

Increasing the exponent is only half of the rule. If the original term is , the antiderivative is not simply . You must divide by the new exponent.

Forgetting the constant of integration

An indefinite integral represents a family of antiderivatives. Always include unless the problem gives a condition that determines the constant.

Using the rule on

The exponent is the one exception. Remember: .

Not rewriting roots or fractions first

Expressions such as and are much easier to integrate after rewriting them using exponents.