Integration by Parts

Date: 04/10/2008 at 11:55:52
From: Arpita
Subject: integration

I'm trying to integrate e^x(x+1)/(x+2)^2 and am stuck.  I tried
letting x+2 = t with 1 = dt/dx but didn't get anywhere.  Can you help me?

Date: 04/11/2008 at 08:56:22
From: Doctor Ali
Subject: Re: integration

Hi Arpita!

Thanks for writing to Dr. Math.  Here's how I'd approach it:

    -
   |    x+1       x
   | --------- * e  dx
   |  (x+2)^2
  -

Write it this way in order to expand the denominator:

    -
   |   x+2-1      x
   | --------- * e  dx
   |  (x+2)^2
  -

Break the fraction into two separate fractions:

    -              -
   |   e^x        |    e^x
   | ------- dx - | --------- dx
   |  (x+2)       |  (x+2)^2
  -              -

Let the first integral be as it is.  Use integration by parts to 
integrate the second one.  We assume that 

  dv = dx/(x+2)^2

We do this way because we know the integral of dv.

          1
  v = - -----
         x+2

Use the main formula of integration by parts:

    -             -
   |  udv = uv - |  vdu
  -             -

Can you continue the process yourself?

Please write back if you still have any difficulties.

- Doctor Ali, The Math Forum
  http://mathforum.org/dr.math/