Integration by parts formula: ∫ u dv = uv − ∫ v du
The integration by parts formula is ∫ u (dv/dx) dx = uv − ∫ v (du/dx) dx.
What each letter means
- \(u\) the part you differentiate (choose it so du/dx is simpler)
- \(\frac{dv}{dx}\) the part you integrate
When to use it
To integrate a product such as x eˣ, x sin x or x ln x, where substitution does not help.
Worked example
Find the exact value of \(\displaystyle\int_1^e x\ln x\,dx\).
- \(u=\ln x\), \(\dfrac{dv}{dx}=x\), so \(\dfrac{du}{dx}=\dfrac1x\), \(v=\tfrac12x^2\)
- \(\left[\tfrac12x^2\ln x-\tfrac14x^2\right]_1^e=\left(\tfrac12e^2-\tfrac14e^2\right)-\left(0-\tfrac14\right)\)
Answer: \(\dfrac{e^2+1}{4}\)
Common mistake
Choosing u badly. With ln x, let u = ln x (you cannot easily integrate it); with x eˣ, let u = x.
On your course
| Course | In the exam |
|---|---|
| Edexcel A Level (9MA0) | In the Edexcel formulae booklet |
| Edexcel International A Level | In Pearson’s IAL formulae booklet |
| AQA or OCR A Level | Check your board’s formula booklet: AQA formulae booklet (PDF) · OCR A Level Maths assessment page |
From our own A Level Maths formula sheets, in our words. Official: Pearson's formulae booklet (PDF) · Pearson's IAL formulae booklet (PDF).
Practise and revise
- Practise A Level Maths practice
- Revise Integration
- Revise IAL P4 integration
- Print Edexcel A Level (9MA0) one-page formula sheet
Questions
What is the integration by parts formula?
The integration by parts formula is ∫ u (dv/dx) dx = uv − ∫ v (du/dx) dx. u: the part you differentiate (choose it so du/dx is simpler); dv/dx: the part you integrate.
Is the integration by parts given in the exam?
Edexcel A Level (9MA0): in the Edexcel formulae booklet. Edexcel International A Level: in Pearson’s IAL formulae booklet. AQA or OCR A Level: check your board’s formula booklet. This comes from our own A Level Maths formula sheets; your teacher has the official booklet.
How do I choose u?
Pick u so that du/dx is simpler. A common order is: logs first, then powers of x, then trig and exponentials last. CBSE textbooks call this ILATE.
Our own wording, examples and card, checked by A Level Math Revision. Not produced or endorsed by Pearson Edexcel, AQA or OCR.