# Integration by Parts

 From: Mike Busfield Date: 27 July 1999 Subject: Integration by Parts We have to do some integration by parts questions, but I missed the class. Please help me do questions like integrating x*cos(x)

### Maths Help suggests:

The integration by parts formula may be written either of these ways:

 or

It doesn't matter which of them you use (we will use the second).

You will have to look at the function you are integrating, and choose:
• A part to differentiate (called u), and
• A part to integrate (called v').

Try to choose them the right way round to make things simpler. With practice,
you will begin to see how to choose your u and v'. Click here for some tips.

For your problem, since  x  becomes very simple when differentiated,
let  u = x, and  v' = cos(x).

Then differentiate u to get u' and integrate v' to get v:

Substitute for u, u', v and v' into the formula:

then you have another integration to do:

Finally, simplify (and don't forget the constant of integration):

You can check your answers by differentiating. In this case, you would
use the product rule of differentiation.