How do you integrate by parts?

When you are faced with an integral which is a product, such as x.cos(x), you may be able to integrate it by parts. The statement of the integration by parts is that:

∫u(dv/dx)dx = uv-∫v(du/dx)dx

So if you have a function of the form u(dv/dx) (such as x.cos(x)​) it can be presented in the above form. In this case u = x and dv/dx = cos(x). 

When doing an intergration by parts it is useful to draw a grid first and work out what v and du/dx are (since we already know u and dv/dx) like so:

u = x

dv/dx = cos(x)

v = sin(x) (this is what you get when you integrate cos(x) as ∫(dv/dx)dx = v)

du/dx = 1 (what you get from differentiating u)

And so now that we have everything we need we can plug things into the equation:

∫x.cos(x)dx = uv-∫v(du/dx)dx = x.sin(x) - ∫1.sin(x)dx

and then to finish we integrate the last bit:

 ∫x.cos(x)dx = x.sin(x)+cos(x)+c (since this is an indefinite integral we must add a constant of integration c).

Answered by Tim W. Maths tutor

4565 Views

See similar Maths A Level tutors

Related Maths A Level answers

All answers ▸

Show that x^2 - 6x + 11> 0 for all values of x


how do you differentiate tan(x)


Integrate (x^2 +2)(2x-6) with respect to x.


Find the equation of the tangent to the curve y=x^3-4x^2+2 at the point (3,-7)


We're here to help

contact us iconContact usWhatsapp logoMessage us on Whatsapptelephone icon+44 (0) 203 773 6020
Facebook logoInstagram logoLinkedIn logo

© MyTutorWeb Ltd 2013–2024

Terms & Conditions|Privacy Policy