# Integration by substitution

Replace a tangled part with a new letter u so the integral becomes simple. Change dx into du too, and change limits if the integral is definite.

Maths (university) · Part 05: Integration · Card 7 of 10 · Last updated 2026-10-19

## Analogy

Renaming a long word to one letter while you work, then translating back.

## Example

**Integral of 2x (x^2 + 1)^3 dx**

Let u = x^2 + 1, so du = 2x dx.
Integral of u^3 du = u^4/4.
Back: (x^2 + 1)^4 / 4 + C.
Differentiate to check.

## In practice

The reverse of the chain rule. Look for a function whose derivative also appears.

## Practice

### Flashcards

- **What substitution suits the integral of 2x (x^2 + 1)^3 dx?** u = x^2 + 1, so du = 2x dx.

## Questions people ask

### How do I choose u in a substitution?

Choose the inside of a bracket, root or exponent, especially if its derivative appears elsewhere in the integral. In 2x times (x squared + 1) cubed, u = x squared + 1 works because du = 2x dx matches the extra factor exactly.

### What happens to the limits in a definite integral?

If you change variable, change the limits to match. For u = x squared + 1 with x from 0 to 1, the new limits are u = 1 and u = 2. Then you do not need to switch back to x at the end.

## Related concepts

- [Integration by parts](https://studyvisually.com/learn/science/maths-university/maths-university-part-6/maths-university-part-6-concept-1)
- [Choosing u and dv](https://studyvisually.com/learn/science/maths-university/maths-university-part-6/maths-university-part-6-concept-2)
- [Partial fractions](https://studyvisually.com/learn/science/maths-university/maths-university-part-6/maths-university-part-6-concept-3)
- [Trigonometric integrals](https://studyvisually.com/learn/science/maths-university/maths-university-part-6/maths-university-part-6-concept-4)

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