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 int(u*v') = uv - int(v*u')

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

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:
u' = 1  and  v = sin(x)

Substitute for u, u', v and v' into the formula:
Integral = x*sin(x) - int(cos(x).dx)

then you have another integration to do:
Integral = x*sin(x) - sin(x)

Finally, simplify (and don't forget the constant of integration):
Integral = sin(x)*(x - 1) + c

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

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