Power Rule for Integration
DRAFT
Derivatives
First let us 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
"multiply by power
then reduce power by 1"
The reverse is:
"add 1 to power
then divide by the new power"
Power Rule for Integration
We can simply work the derivative rule backwards
Just as the Power Rule for Derivatives helps us find the derivative of functions like xn, the Power Rule for Integration helps us find the integral, or antiderivative, of similar functions. Integration is like reversing the process of differentiation.
The 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.
Understanding Through Example
Let's use a real-world scenario to understand how this rule is applied.
Example: Finding the integral of x2:
When a car accelerates uniformly, its position as a function of time could be similar to x2. Suppose we want to find the position from this velocity function using integration. The function we have is x2, where n = 2. Here's how we apply the Power Rule:
| The integral of x2 | = | 1(2+1)x2+1 + C |
| = | 13x3 + C |
Answer: the integral of x2 is 13x3 + C
Thus, at any time t, the position of the car could be expressed in terms of x cubed. Don't forget the '+ C' for any additional constants that may affect initial positioning!
How to Remember
"multiply by power
then reduce power by 1"
A Short Table
Here is the Power Rule with some sample values. See the pattern?
| f | f’(xn) = nx(n−1) | f’ |
|---|---|---|
| x | 1x(1−1) = x0 | 1 |
| x2 | 2x(2−1) = 2x1 | 2x |
| x3 | 3x(3−1) = 3x2 | 3x2 |
| x4 | 4x(4−1) = 4x3 | 4x3 |
| etc... | ||
| And for negative exponents: | ||
| x-1 | −1x(−1−1) = −x-2 | −x-2 |
| x-2 | −2x(−2−1) = −2x-3 | −2x-3 |
| x-3 | −3x(−3−1) = −3x-4 | −3x-4 |
| etc... | ||