# Power Rule in Integration

How to solve the power rule in integration problems: formula, examples, and their solutions.

## Formula

The integral of *x*^{n} *dx* is equal to

(1/[*n* + 1])*x*^{n + 1} + *C*.

First change the exponent to *n* + 1.

Then multiply the reciprocal of *n* + 1, 1/(*n* + 1),

in front of *x*^{n + 1}.

This is true because

[ (1/[*n* + 1])*x*^{n + 1} + *C* ]' = ([*n* + 1] / [*n* + 1])*x*^{n}

= *x*^{n}.

Power rule in differentiation (Part 1)

## Example 1

Change the exponent to, 3 + 1, 4.

Then multiply the reciprocal of 4, 1/4.

So (given) = (1/4)*x*^{4} + *C*.

## Example 2

√*x* = *x*^{1/2}

Rational exponents

Change the exponent to, 1/2 + 1, 3/2.

Then multiply the reciprocal of 3/2, 2/3.

So (given) = (2/3)*x*^{3/2} + *C*.

*x*^{3/2} = *x*^{1 + 1/2}

= *x*√*x*

So (given) = (2/3)*x*√*x* + *C*.