Reverse Power Rule for Integrals: Formula, Examples and n = −1
The reverse power rule: the integral of x^n is x^(n+1)/(n+1) + C for n ≠ −1. Formula, examples with roots and negative powers, and the ln|x| exception.
Integration rules Updated 3 min read
At a glance
- Applies to
- Any power of x with a constant exponent n \ne -1
- In words
- Add one to the exponent, then divide by the new exponent
- Exception
- \int x^{-1}\,dx = \ln\lvert x\rvert + C
- Check
- Differentiate the answer with the power rule; you should get x^n back
- Related rules
- Power rule, linearity of the integral, u-substitution
On this page
The reverse power rule finds the antiderivative of any power of x. It is the power rule run backwards: instead of lowering the exponent and multiplying, you raise the exponent and divide.
The reverse power rule formula
For any constant exponent n \ne -1:
Two moves: add one to the exponent, then divide by the new exponent. The +C is there because every function has infinitely many antiderivatives that differ by a constant.
You can always check the answer by differentiating it:
Examples
| Integrand | Rewrite | Antiderivative |
|---|---|---|
| x^4 | x^4 | \frac{x^5}{5} + C |
| x | x^1 | \frac{x^2}{2} + C |
| 5 | 5x^0 | 5x + C |
| \sqrt{x} | x^{1/2} | \frac{2}{3}x^{3/2} + C |
| \frac{1}{x^3} | x^{-3} | -\frac{1}{2x^2} + C |
| \frac{1}{\sqrt{x}} | x^{-1/2} | 2\sqrt{x} + C |
| \frac{1}{x} | x^{-1} | \ln\lvert x\rvert + C (exception) |
Example: a polynomial. Combine the rule with linearity and integrate term by term:
Example: a negative power. Rewrite \frac{4}{x^5} as 4x^{-5}:
The exponent goes from -5 up to -4. Adding one to a negative number makes it less negative.
Example: a fractional power. For \int \sqrt[3]{x}\,dx, write x^{1/3}. The new exponent is \frac{4}{3}, and dividing by \frac{4}{3} means multiplying by \frac{3}{4}:
The n = −1 exception
When n = -1 the formula asks you to divide by zero, so it breaks. The antiderivative of x^{-1} is the natural logarithm:
The absolute value makes the answer valid for negative x too. This is why \frac{1}{x} gets its own row on every integral table.
When to use the reverse power rule
Use it when the integrand is a sum of terms of the form kx^n, possibly after rewriting:
- polynomials, one term at a time;
- roots, after converting to fractional exponents;
- terms with x in the denominator, after converting to negative exponents;
- products or quotients you can expand or split first, such as \frac{x^3 + 1}{x} = x^2 + x^{-1}.
If the base is something other than plain x, such as (3x-2)^5, you need u-substitution. If the variable is in the exponent, as in 2^x, you need the exponential rule \int a^x\,dx = \frac{a^x}{\ln a} + C.
Common mistakes
- Dividing by the old exponent. \int x^3\,dx = \frac{x^4}{4} + C, not \frac{x^4}{3} + C.
- Applying it to products. \int x(x+1)\,dx is not \frac{x^2}{2}\cdot\left(\frac{x^2}{2}+x\right). Expand to x^2 + x first.
- Using it on \frac{1}{x}. \int x^{-1}\,dx is \ln\lvert x\rvert + C, not \frac{x^0}{0}.
- Forgetting +C. An indefinite integral is a whole family of functions.
- Treating a power of a binomial like a power of x. \int (2x+1)^3\,dx \ne \frac{(2x+1)^4}{4} + C. Differentiating that answer gives an extra factor of 2.
For the bigger picture of antiderivatives, including the trig and exponential rules, read the full guide to antiderivatives.
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The reverse power rule
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