How do you integrate by parts?

  • Google+ icon
  • LinkedIn icon
  • 664 views

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,...

About the author

is an online A Level Maths tutor who has applied to tutor with MyTutor studying at Durham University

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

95% of our customers rate us

Browse tutors

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

mtw:mercury1:status:ok