Integration Power Rule

July 9, 2026

Quick overview of the integration power rule.

Main Idea

The integration power rule is

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

as long as

\[n\ne -1.\]

This mainly reverses the derivative power rule, where we add 1 to the power and divide by that new power.

Watch this section: introduction at 0:00.

A simple example of this rule in action follows

\[\int x^5\,dx.\]

Here the exponent is $5$. Add $1$ to the exponent:

\[5+1=6.\]

Then divide by the same new exponent:

\[\int x^5\,dx=\frac{x^6}{6}+C.\]

Watch this section: example 1 at 0:45.

Let’s move onto examples where we have more than one expression/term added together within our integral. How do we apply this rule then?

The idea is:

\[\int \left(\text{term 1}+\text{term 2}+\text{term 3}\right)dx =\int (\text{term 1})dx+\int (\text{term 2})dx+\int (\text{term 3})dx.\]

You treat each term as an individual integral and apply the power rule for each term.

Watch this section: example 2 at 1:15.

What happens when the power is $-1$? ($n=-1$)

Well

\[n+1=-1+1=0.\]

The term $x^{-1}$ is the same as $\frac{1}{x}$, and it uses its own antiderivative:

\[\int \frac{1}{x}\,dx=\int x^{-1}\,dx=\ln\lvert x\rvert+C.\]

So the power rule handles powers like $x^5$, $x^4$, $x^2$, and $x^{-2}$, but the special power $x^{-1}$ becomes a natural logarithm.

Section	Main idea	Video
Introduction	The rule adds $1$ to the exponent and divides by the new exponent.	0:00
Example 1	Use the rule on $\int x^5\,dx$.	0:45
Example 2	Split a longer expression into terms before integrating.	1:15
Exception	When $n=-1$, use $\int \frac{1}{x}\,dx=\ln\lvert x\rvert+C$.	3:20

This rule is the integration-side partner to the derivative power rule. For derivative practice, use the power rule derivative examples.

Because the answers are indefinite integrals, every result needs $+C$. For the reason behind that constant, review the constant of integration notes.