Power Rule for Integration
Antiderivative of a power function (n≠-1).
Formula
Variables
Example
∫x²dx = x³/3 + C
How it works
This rule tells you how to find an antiderivative of a power of x: to reverse differentiation, increase the exponent by 1 and divide by the new exponent. It works because differentiating x^(n+1)/(n+1) gives x^n by the power rule, as long as n is not -1. The special case n = -1 is excluded because it would require dividing by zero, and that case instead leads to ln|x|.
When to use it
Use this when integrating simple polynomial terms like x^5, x^(1/2), or x^-3. It is especially useful when finding areas, displacement from velocity, or antiderivatives in basic calculus problems.
Worked example, step by step
Find ∫x^4 dx Step 1: Identify the exponent: n = 4 Step 2: Add 1 to the exponent: 4 + 1 = 5 Step 3: Divide by the new exponent: x^5 / 5 Step 4: Add the constant of integration Final answer: ∫x^4 dx = x^5/5 + C
Common mistakes to avoid
- Forgetting to add 1 to the exponent before dividing
- Using the rule when n = -1 instead of writing ln|x| + C
- Omitting the + C when writing the antiderivative
Did You Know?
The power rule for integration fails for 1/x, and that single exception is exactly where the natural logarithm comes from.
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