# Standard integrals

Reverse the standard derivatives: cos x integrates to sin x, sin x to -cos x, e^x to e^x and 1/x to ln|x|, each plus C.

Maths (university) · Part 05: Integration · Card 6 of 10 · Last updated 2026-10-19

## Analogy

A reverse times table, memorised so you can recognise patterns quickly.

## Example

**Integral of sin x from 0 to pi**

Antiderivative: -cos x.
At pi: -cos(pi) = 1.
At 0: -cos(0) = -1.
Result: 1 - (-1) = 2.

## In practice

Every harder technique ends by reducing the problem to one of these.

## Practice

### Flashcards

- **What is the integral of sin x from 0 to pi?** 2.
- **What is the integral of 1/x dx?** ln|x| + C.

## Questions people ask

### Why does the integral of sin x have a minus sign?

Because the derivative of cos x is minus sin x. To get plus sin x back, you need minus cos x. You can always check by differentiating your answer: the derivative of minus cos x is sin x, which matches the original.

### Why is the integral of 1/x written with absolute value signs?

The natural logarithm only accepts positive inputs, but 1/x exists for negative x too. Writing ln of absolute x covers both sides of zero, and its derivative is still 1/x. Over any interval that crosses zero, the integral does not exist.

## Related concepts

- [Antiderivative](https://studyvisually.com/learn/science/maths-university/maths-university-part-5/maths-university-part-5-concept-1)
- [Power rule for integrals](https://studyvisually.com/learn/science/maths-university/maths-university-part-5/maths-university-part-5-concept-2)
- [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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