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### 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/
```
Associated Topics:
High School Calculus

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