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

5704 Views

See similar Maths A Level tutors

Related Maths A Level answers

All answers ▸

Prove or disprove the following statement: ‘No cube of an integer has 2 as its units digit.’


C1 - Simplifying a fraction that has a root on the denominator


Use the substitution u=4x-1 to find the exact value of 1/4<int<1/2 ((5-2x)(4x-1)^1/3)dx


Find the indefinite integral of Ln(x)


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 2013–2025

Terms & Conditions|Privacy Policy
Cookie Preferences