# 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.

Source: https://practicecalculus.com/rules/reverse-power-rule
Updated: 2026-11-03

$$ \int x^n\,dx = \frac{x^{n+1}}{n+1} + C, \quad n \ne -1 $$

## 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

## Key takeaways

- The antiderivative of $x^n$ is $\frac{x^{n+1}}{n+1} + C$ for every constant $n \ne -1$.
- Rewrite roots and fractions as powers first: $\sqrt{x} = x^{1/2}$ and $\frac{1}{x^2} = x^{-2}$.
- The case $n = -1$ is special: $\int \frac{1}{x}\,dx = \ln\lvert x\rvert + C$.
- Every indefinite integral needs a $+ C$.

The **reverse power rule** finds the antiderivative of any power of $x$. It is the [power rule](/rules/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 $n \ne -1$:

$$\int x^n\,dx = \frac{x^{n+1}}{n+1} + C$$

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:

$$\frac{d}{dx}\left(\frac{x^{n+1}}{n+1}\right) = \frac{(n+1)x^{n}}{n+1} = x^n$$

```graph
title: 'Graph of y = x² and its antiderivatives y = x³/3 + C'
caption: 'Each dashed curve is $\frac{x^3}{3} + C$ for a different $C$. They are vertical shifts of one another, and all have slope $x^2$ at every $x$.'
x: [-2.5, 2.5]
y: [-4, 5]
functions:
  - expr: x^2
    label: y = x²
  - expr: x^3/3
    label: y = x³/3
    dashed: true
  - expr: x^3/3 + 2
    label: y = x³/3 + 2
    dashed: true
  - expr: x^3/3 - 2
    label: y = x³/3 − 2
    dashed: true
```

## 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](/rules/integral-linearity-rules) and integrate term by term:

$$\int \left(6x^2 - 4x + 3\right)dx = 2x^3 - 2x^2 + 3x + C$$

**Example: a negative power.** Rewrite $\frac{4}{x^5}$ as $4x^{-5}$:

$$\int 4x^{-5}\,dx = 4 \cdot \frac{x^{-4}}{-4} + C = -\frac{1}{x^4} + C$$

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 $\frac{3}{4}$:

$$\int x^{1/3}\,dx = \frac{3}{4}x^{4/3} + C$$

## 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:

$$\int \frac{1}{x}\,dx = \ln\lvert x\rvert + C$$

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](/rules/u-substitution-rule). 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

1. **Dividing by the old exponent.** $\int x^3\,dx = \frac{x^4}{4} + C$, not $\frac{x^4}{3} + C$.
2. **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.
3. **Using it on $\frac{1}{x}$.** $\int x^{-1}\,dx$ is $\ln\lvert x\rvert + C$, not $\frac{x^0}{0}$.
4. **Forgetting $+C$.** An indefinite integral is a whole family of functions.
5. **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](/blog/antiderivatives).

## Frequently asked questions

### What is the reverse power rule?

It is the integration version of the power rule: $\int x^n\,dx = \frac{x^{n+1}}{n+1} + C$ for $n \ne -1$. You raise the exponent by one and divide by the new exponent.

### Why does the reverse power rule not work for 1/x?

For $n = -1$ the formula would divide by $n + 1 = 0$. The antiderivative of $\frac{1}{x}$ is instead $\ln\lvert x\rvert + C$.

### How do you integrate a square root with the power rule?

Write $\sqrt{x}$ as $x^{1/2}$. Then $\int x^{1/2}\,dx = \frac{x^{3/2}}{3/2} + C = \frac{2}{3}x^{3/2} + C$.

### What is the integral of a constant?

A constant $k$ is $kx^0$, so the rule gives $\int k\,dx = kx + C$.

### Can I use the reverse power rule on (2x + 1)^3?

Not directly, because the base is not just $x$. Use u-substitution with $u = 2x + 1$, which gives $\frac{(2x+1)^4}{8} + C$, or expand the cube first.