Power Rule for Integration

Derivatives

First let's look at the Power Rule for derivatives, one of the most commonly used rules in Calculus:

The derivative of xn is nx(n−1)

Example: What is the derivative of x2 ?

For x2 we use the Power Rule with n=2:

The derivative of   x2   =   2x(2−1)
    =   2x1
    =   2x

Answer: the derivative of x2 is 2x

Reverse the Rule

So for derivatives we have:

power rule x^3 3x^2
"multiply by power
then reduce power by 1"

And for integration we reverse the rule:

power rule x^3 3x^2
"add 1 to power
then divide by the new power"

Power Rule for Integration

The integral of xn is 1(n+1)x(n+1) + C, for n ≠ -1

Here, C is the constant of integration. We add this constant because when we differentiate any constant, it becomes zero. Therefore, when integrating, we must account for the unknown constant that might have been there originally.

Example: Finding the integral of x2:

The integral of   x2   =   1(2+1)x2+1 + C
    =   13x3 + C

Answer: ∫ x2 = x3 3+ C

Example: What is ∫√x dx ?

√x is also x0.5

∫x0.5 dx = x1.51.5 + C

= 23x1.5 + C

How to Remember

power rule x^3 3x^2
"add 1 to power
then divide by the new power"

A Short Table

Here is the Power Rule for Integration with some sample values. See the pattern?

f(x) ∫ f(x) dx
x 12x2 + C
x2 13x3 + C
x3 14x4 + C
x4 15x5 + C
and so on...  
For negative exponents (except −1):
x−2 1−1x−1 + C
= −x−1 + C
x−3 1−2x−2 + C
and so on...  
Power Rule Derivative Rules Calculus Index