Power Rule (Integral)
Antiderivative of a power function.
Formula
Variables
Example
∫ x² dx = x³/3 + C.
How it works
This rule tells you how to reverse the process of differentiating a power of x. It works because differentiation of x^(n+1) produces (n+1)x^n, so dividing by (n+1) undoes that effect; the extra +C is needed because any constant disappears when differentiated. The rule does not apply when n = -1, because x^-1 integrates to ln|x| rather than another power.
When to use it
Use this when integrating a single power of x, or a polynomial term-by-term after splitting the sum into separate powers. It is common in physics, engineering, and economics whenever a rate that depends on x^n needs to be accumulated into total change.
Worked example, step by step
Find ∫ 5x^4 dx Step 1: Keep the constant 5 outside the integral: 5∫ x^4 dx Step 2: Apply the power rule: 5 · (x^5/5) + C Step 3: Simplify: x^5 + C Final answer: ∫ 5x^4 dx = x^5 + C
Common mistakes to avoid
- Forgetting to add 1 to the exponent before dividing
- Using the rule when n = -1 instead of switching to ln|x|
- Leaving out the constant of integration C
Did You Know?
The +C on every indefinite integral reminds us that infinitely many curves share the same slope.
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IntermediateDerivative of a power function.
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IntermediateDerivative of a composite function.
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IntermediateDerivative of the cosine function.