Integration by parts formula: ∫ f g dx = f ∫ g dx − ∫ (f′ ∫ g dx) dx
The integration by parts formula is ∫ f(x) g(x) dx = f(x) ∫ g(x) dx − ∫ [f′(x) ∫ g(x) dx] dx.
What each letter means
- \(f(x)\) the first function, chosen by ILATE (you differentiate it)
- \(g(x)\) the second function (you integrate it)
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
Evaluate \(\displaystyle\int x\cos x\,dx\).
- By ILATE, first function \(f=x\), second \(g=\cos x\)
- \(x\sin x-\displaystyle\int1\cdot\sin x\,dx\)
Answer: \(x\sin x+\cos x+C\)
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 |
|---|---|
| Class 12 | Learn it: CBSE gives no formula booklet |
From our own CBSE Maths formula sheets, in our words.
Practise and revise
- Practise Class 12 Integrals MCQs
- Revise Class 12 Integrals
- Print Class 12 one-page formula sheet
![Integration by parts formula card: ∫ f(x) g(x) dx = f(x) ∫ g(x) dx − ∫ [f′(x) ∫ g(x) dx] dx. CBSE Math Revision](/cbse-frontend/img/formula-cards/cbse-maths-integration-by-parts-formula-card.png)
Questions
What is the integration by parts formula?
The integration by parts formula is ∫ f(x) g(x) dx = f(x) ∫ g(x) dx − ∫ [f′(x) ∫ g(x) dx] dx. f(x): the first function, chosen by ILATE (you differentiate it); g(x): the second function (you integrate it).
Is the integration by parts given in the exam?
Class 12: learn it: CBSE gives no formula booklet. This comes from our own CBSE 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 CBSE Math Revision. Not produced or endorsed by CBSE or NCERT.