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...

1 year ago

Answered by Harry, an A Level Maths tutor with MyTutor

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


£20 /hr

Jessica M.

Degree: Medicine (Bachelors) - Manchester University

Subjects offered: Maths, History+ 4 more

-Medical School Preparation-

“Hello! I'm currently a fourth year medical student at the University of Manchester. I have always loved science and I have lots of experience tutoring younger students in school. I personally learn things much quicker in an interactiv...”

MyTutor guarantee

£20 /hr

Alex A.

Degree: Physics (Masters) - Manchester University

Subjects offered: Maths, Science+ 4 more

-Personal Statements-

“Hi, I'm currently studying for a Masters Degree in Physics at the University of Manchester. I'm keen on all things Maths and Science related, as well as having interests ranging from Rock Climbing to Acting. I've always thought that a ...”

MyTutor guarantee

£20 /hr

Thomas P.

Degree: Masters in Mathematics (Masters) - Oxford, Oriel College University

Subjects offered: Maths, Further Mathematics

Further Mathematics

“About Me: I am a Maths student at Oxford University. Whilst I love learning about, talking about, and generally engaging with Mathematics, that's not all I do! Outside of studying I enjoy board games, and I play squash for the Univers...”

About the author

£22 /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 I know when to differentiate using the chain rule? And how do I do this?

Differentiate y=(4x - 5)^5 by using the chain rule.

Differentiate y^3 + 3y^2 + 5

How do I integrate log(x) or ln(x)?

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