How do you integrate by parts?

This is one of the trickier methods of integration, and it requires some practise. The basic idea is to split a function which would be difficult to integrate into two parts. Differentiating one part and integrating the other will then lead to a function which is much easier to integrate.

The formula is that the integral of u dv = uv - the integral of v du. It is best demonstrated with an example:

Let's integrate f(x) = xcos(x)

We can see that x will disppear if we differentiate it, so let's set x = u and cos(x) = dv.

Differentiating u and integrating dv then gives du = 1 and v = sin(x)

Now we substitute these into the formula: xsin(x) - integral of sin(x)

Sin(x) is easy to integrate, it is just -cos(x). Now we have our answer! The integral of xcos(x) = xsin(x) + cos(x) + c, where c is our unknown (and always necessary!) constant of integration.

Harry M. GCSE Maths tutor, A Level Maths tutor, GCSE Physics tutor, A...

8 months ago

Answered by Harry, an A Level Maths tutor with MyTutor

Still stuck? Get one-to-one help from a personally interviewed subject specialist


PremiumRebecca V. A Level Maths tutor, 13 plus  Maths tutor, GCSE Maths tuto...
View profile
£20 /hr

Rebecca V.

Degree: MA Logic and Philosophy of Mathematics (Masters) - Bristol University

Subjects offered: Maths, Physics+ 1 more

Further Mathematics

“Hi, I'm Rebecca, I'm 23 and I'm currently taking a year to myself just to study, doing my masters in Logic and Philosophy of Maths. To me, mathematics has always been black and white, right or wrong whilst philosophy is all about thin...”

Seb G. 11 Plus Maths tutor, 13 plus  Maths tutor, A Level Maths tutor...
View profile
£20 /hr

Seb G.

Degree: Mathematics (Masters) - Bath University

Subjects offered: Maths


“Me: I'm a second year maths student at the University of Bath. I fell in love with maths after a rocky start with the subject, so believe that I can help people regardless of whether they like maths or not.  I've been tutoring maths l...”

MyTutor guarantee

Annabel W. GCSE Maths tutor, A Level Maths tutor, GCSE Chemistry tutor
View profile
£20 /hr

Annabel W.

Degree: Mathematics (Masters) - Durham University

Subjects offered: Maths, Chemistry


“Hi! I'm Annie, a first year Mathematics student studying a Durham University. If you're looking for support in your studies in Maths or Chemistry, I'm here to help! I'm very enthusiastic about helping others to learn and progress, and ...”

About the author

Harry M. GCSE Maths tutor, A Level Maths tutor, GCSE Physics tutor, A...
View profile
£20 /hr

Harry M.

Degree: Physics (Masters) - Exeter University

Subjects offered: Maths, Physics


“Hey everyone,If you're reading this, then I assume you're stuck on something physics-y or maths-y. If so, thenI can help! I'm a Physics student at Exeter University and I can be your guru. Physics reduces the complex world to somethi...”

You may also like...

Posts by Harry

Explain Newton's laws of motion

Explain Rutherford's atomic model experiment

How do you integrate by parts?

How do you multiply out brackets

Other A Level Maths questions

How do you differentiate X to the power of a?

A pot of water is heated to 100C and then placed in a room at a temperature of 18C. After 5 minutes, the pan temperature falls by 20C. Find the temperature after 10minutes.

How to find out where 2 lines cross/simultaneous equations

Find dy/dx where y=e^(4xtanx)

View A Level Maths tutors


We use cookies to improve our service. By continuing to use this website, we'll assume that you're OK with this. Dismiss