# Power rule for integrals

The integral of x^n dx is x^(n+1) / (n+1) + C, for n not equal to -1. Raise the power by one, then divide by the new power.

Maths (university) · Part 05: Integration · Card 2 of 10 · Last updated 2026-10-19

## Analogy

The power rule for derivatives, run backwards.

## Example

**Integral of 6x^2 - 4x + 1**

6x^2 gives 6x^3/3 = 2x^3.
-4x gives -4x^2/2 = -2x^2.
1 gives x.
Result: 2x^3 - 2x^2 + x + C.

## In practice

Lets you integrate any polynomial one term at a time.

## Practice

### Flashcards

- **State the power rule for integrals.** The integral of x^n dx is x^(n+1)/(n+1) + C, for n not equal to -1.

### Interview questions

**Evaluate the integral of 3x^2 + 2x from 1 to 2 and explain each step.**

- Antiderivative is x^3 + x^2
- Evaluate at 2: 8 + 4 = 12
- Evaluate at 1: 1 + 1 = 2
- Subtract: 12 - 2 = 10
- Fundamental theorem of calculus justifies it

## Questions people ask

### Why does the power rule for integrals fail when n is -1?

Dividing by n + 1 would mean dividing by zero. The integral of 1 over x is a different function: the natural logarithm of absolute x, plus C. This is the one power of x that needs its own rule in the table.

### How do I integrate roots and fractions?

Rewrite them as powers first. The square root of x is x to the one-half, and 1 over x squared is x to the minus two. Then apply the power rule: the integral of x to the one-half is two-thirds x to the three-halves.

## Related concepts

- [Fundamental theorem of calculus](https://studyvisually.com/learn/science/maths-university/maths-university-part-5/maths-university-part-5-concept-5)
- [Antiderivative](https://studyvisually.com/learn/science/maths-university/maths-university-part-5/maths-university-part-5-concept-1)
- [Riemann sum](https://studyvisually.com/learn/science/maths-university/maths-university-part-5/maths-university-part-5-concept-3)
- [Definite integral](https://studyvisually.com/learn/science/maths-university/maths-university-part-5/maths-university-part-5-concept-4)

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