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 \(\displaystyle\int xe^{2x}\,dx\).
- \(u=x\), \(\dfrac{dv}{dx}=e^{2x}\), so \(\dfrac{du}{dx}=1\), \(v=\tfrac12e^{2x}\)
- \(\tfrac12xe^{2x}-\displaystyle\int\tfrac12e^{2x}\,dx\)
Answer: \(\tfrac12xe^{2x}-\tfrac14e^{2x}+C\)
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 |
|---|---|
| AA HL | In the IB formula booklet |
From our own IB Maths formula sheets, in our words. The IB gives its booklet to schools, so we checked against our own copy: your teacher has the official one. Official: IB DP Mathematics page.
Practise and revise
- Practise AA HL advanced integration questions
- Revise Integration by parts (AA HL)
- Print AA HL 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?
AA HL: in the IB formula booklet. This comes from our own IB 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 IB Math Revision. Not produced or endorsed by the International Baccalaureate Organization.