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).

Timothy W. GCSE Maths tutor, A Level Maths tutor

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