AP Calculus AB and BC glossary

Reverse power rule

Also called: Power rule for integrals, Integral power rule

The reverse power rule integrates a power of x by adding one to the exponent and dividing by that new exponent, then adding a constant. It works for every exponent except negative one. That one excluded case, one over x, integrates instead to the natural log of the absolute value of x.

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

It is the power rule read backwards. Where differentiating drops the exponent by one and multiplies, integrating raises the exponent by one and divides. You can confirm any answer by differentiating it, which turns $\frac{x^{n+1}}{n+1}$ straight back into $x^n$ .

Its reach is wider than it first looks once you rewrite roots and reciprocals as powers. Here $\sqrt{x}$ becomes $x^{1/2}$ and $\frac{1}{x^3}$ becomes $x^{-3}$ , so both integrate by the same step: $\int \sqrt{x}\,dx = \frac{2}{3}x^{3/2} + C$ and $\int x^{-3}\,dx = -\frac{1}{2}x^{-2} + C$ .

The mistake

Forcing the rule onto $\frac{1}{x}$ . Since $n = -1$ makes $n+1 = 0$ , the pattern gives $\frac{x^0}{0}$ , which is undefined. This one exponent breaks off from the family: $\int \frac{1}{x}\,dx = \ln|x| + C$ , with the absolute value that keeps it defined for negative $x$ .

Appears in: Unit 6: Integration and Accumulation