# Choosing u and dv

A good choice makes the new integral simpler than the old one. Logarithms and inverse trig functions go in u. Exponentials and sines are usually dv.

Maths (university) · Part 06: Integration techniques and uses · Card 2 of 9 · Last updated 2026-10-19

## Analogy

Choosing which piece of flat-pack furniture to assemble first so the rest fits easily.

## Example

**Integral of x ln x dx**

u = ln x, dv = x dx.
du = dx/x, v = x^2/2.
(x^2/2) ln x - integral of x/2 dx.
Result: (x^2/2) ln x - x^2/4 + C.

## In practice

A wrong choice makes the integral harder, which tells you to swap the roles.

## Practice

### Flashcards

- **Which function is usually u: ln x or x?** ln x, because it simplifies when differentiated.

### Interview questions

**Evaluate the integral of x e^x dx and explain how you chose u and dv.**

- Let u = x so du = dx
- Let dv = e^x dx so v = e^x
- Formula gives x e^x minus integral of e^x
- Answer x e^x - e^x + C
- Differentiate to check

## Questions people ask

### Why can't I integrate ln x directly by a table?

The natural log has no simple antiderivative from the standard list. But by parts works with u = ln x and dv = dx. This gives x ln x minus x, plus C. Check by differentiating: the result is ln x, as needed.

### What if integrating by parts makes things worse?

Swap your choice of u and dv. If the new integral looks harder than the original, it was a poor choice. Sometimes you need to apply the method twice, as with x squared times e to the x, lowering the power of x each time.

## Related concepts

- [Integration by parts](https://studyvisually.com/learn/science/maths-university/maths-university-part-6/maths-university-part-6-concept-1)
- [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)
- [Integration by substitution](https://studyvisually.com/learn/science/maths-university/maths-university-part-5/maths-university-part-5-concept-7)

Web page: https://studyvisually.com/learn/science/maths-university/maths-university-part-6/maths-university-part-6-concept-2
Study this card in the app: https://studyvisually.com/app/learn?category=maths-university&card=maths-university-part-6-concept-2