How do you integrate by parts?

This is one of the trickiest methods of calculus on the course, but it's important to know, and is very doable if you set up the problem right and remember the steps. 

Integration by parts works when you have to integrate a function of the type f=u(dv/dx). All you have to remember is that, and the formula(dv/dx) dx = uv - ∫ (du/dx) dx

Ok, let's try an example. 

Say you're asked to integrate xsin(x). 

I find it makes it easiest to write out all the things I need for the formula before I plug them in. 

We'll choose x to be u, because differentiating x makes it more simple, while differentiating sin(x) doesn't really help that much. You always choose u to be the part that comes out simplest when differentiated.

So:                                                            u = x

Then, by differentiating,                du/d= 1

and also:                                               dv/dx = sin(x)

Then, integrating to find v,             v = -cos(x). 

Now, all we have to do is plug that back into the formula from earlier:

             ∫ xsin(x) dx = -xcos(x) - ∫ -cos(x) (1) dx.

Which is way easier! Integrating cos(x) gives sin(x) + c (always remember c!), so we end up with

             ∫ xsin(x) dx = -xcos(x) + sin(x) + c.

And that's your answer!

Isaac E. GCSE Physics tutor, A Level Physics tutor, GCSE Maths tutor,...

9 months ago

Answered by Isaac, who has applied to tutor A Level Maths with MyTutor

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


£24 /hr

Alex B.

Degree: Physics (Masters) - Durham University

Subjects offered: Maths, Physics


“About me: I’m Alex and I have just completed my first year studying physics at Durham University, with a first. I have a passion for mathematics and how it can be applied to solve problems in physics. Throughout my A levels, other stu...”

£20 /hr

Isabel R.

Degree: Mathematics (Bachelors) - Manchester University

Subjects offered: Maths, Physics+ 1 more

Further Mathematics

“I am currently a first year studying Mathematics at the University of Manchester-so A Levels and GCSEs are still fresh in my mind when it comes to remembering how I learnt the material myself. In school I mentored GCSE students in Mat...”

£22 /hr

Jonathan P.

Degree: Physics (Masters) - Durham University

Subjects offered: Maths, Physics


“About Me: I am a Physics student at Durham University. My passion is physics, and by association maths, I will aim to instil asimilar passion in students during my tutorials. I am a highly patient and approachable individual.  I have...”

About the author

£20 /hr

Isaac E.

Degree: Physics (Masters) - Durham University

Subjects offered: Maths, Physics


“Top tutor from the renowned Russell university group, ready to help you improve your grades.”

MyTutor guarantee

You may also like...

Other A Level Maths questions

How do I know which method of integration to use?

Differentiate x^2 + 2x + 7

A rollercoaster stops at a point with GPE of 10kJ and then travels down a frictionless slope reaching a speed of 10 m/s at ground level. After this, what length of horizontal track (friction coefficient = 0.5) is needed to bring the rollercoaster to rest?

How do I find the maximum/minimum of a function?

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