Power Rule in Integration

Power Rule in Integration

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

Formula

(integral of x^n dx) = [1/(n + 1)]*x^(n + 1) + C

The integral of xn dx is equal to
(1/[n + 1])xn + 1 + C.

First change the exponent to n + 1.
Then multiply the reciprocal of n + 1, 1/(n + 1),
in front of xn + 1.

This is true because
[ (1/[n + 1])xn + 1 + C ]' = ([n + 1] / [n + 1])xn
= xn.

Power rule in differentiation (Part 1)

Example 1

Find the given indefinite integral. The integral of x^3 dx

Change the exponent to, 3 + 1, 4.
Then multiply the reciprocal of 4, 1/4.

So (given) = (1/4)x4 + C.

Example 2

Find the given indefinite integral. The integral of sqrt(x) dx

x = x1/2

Rational exponents

Change the exponent to, 1/2 + 1, 3/2.
Then multiply the reciprocal of 3/2, 2/3.

So (given) = (2/3)x3/2 + C.

x3/2 = x1 + 1/2
= xx

So (given) = (2/3)xx + C.