Integration Power Rule
This page follows the JoeCMath video Integration Power Rule Is Easier Than You Think. The video keeps the rule simple: add $1$ to the exponent, divide by the new exponent, and remember $+C$ for an indefinite integral.
Main Idea
The rule shown in the video is
\[\int x^n\,dx=\frac{x^{n+1}}{n+1}+C,\]as long as
\[n\ne -1.\]The thumbnail and opening frames both emphasize the same motion: $n$ becomes $n+1$, and the new exponent also becomes the denominator.
Watch this section: introduction at 0:00.
Example 1: A Single Power Of $x$
The first worked example in the video is
\[\int x^5\,dx.\]Here the exponent is $5$. Add $1$ to the exponent:
\[5+1=6.\]Then divide by the same new exponent:
\[\int x^5\,dx=\frac{x^6}{6}+C.\]This is the cleanest version of the rule because the integrand is already a single power of $x$.
Watch this section: example 1 at 0:45.
Example 2: Use The Rule Term By Term
The second example moves from one power to a longer expression. The video treats the integral as a sum of smaller integrals, then applies the same power-rule move to each piece that is a power of $x$.
The idea is:
\[\int \left(\text{term 1}+\text{term 2}+\text{term 3}\right)\,dx =\int \text{term 1}\,dx+\int \text{term 2}\,dx+\int \text{term 3}\,dx.\]For a constant multiple, the coefficient stays in front while the power of $x$ changes:
\[\int a x^n\,dx =a\cdot\frac{x^{n+1}}{n+1}+C, \qquad n\ne -1.\]That is the main habit from the second example: do not try to integrate the whole expression in one jump. Split it into terms, handle each term, then combine the antiderivatives.
Watch this section: example 2 at 1:15.
The n Equals Negative 1 Exception
Near the end, the video returns to the condition $n\ne -1$. That condition matters because substituting $n=-1$ into the power rule would create a zero denominator:
\[n+1=-1+1=0.\]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.
Watch this section: the $n=-1$ case at 3:20.
Timestamp Guide
Related Calculus Work
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.
You can also continue with u-substitution or the full JoeCMath integration playlist.