1. ## Three integration problems

Hi, I've got an integration question I need to answer as part of a assignment and I'm really having trouble making any sort of start on it. I'm not just looking for the answer, I'm hoping to be able to understand how it's reached. From what I've read so far, I think I'm supposed to be integrating by parts and finding an appropriate value for u, but that's about all I can work out so far. The question is:

Find
$
\int \frac{x^3}{(x + 1)^2} dx
$

2. Hello.

Originally Posted by drew.walker
Hi, I've got an integration question I need to answer as part of a assignment and I'm really having trouble making any sort of start on it. I'm not just looking for the answer, I'm hoping to be able to understand how it's reached. From what I've read so far, I think I'm supposed to be integrating by parts and finding an appropriate value for u, but that's about all I can work out so far. The question is:

Find the integral of
$
x^3/(x + 1)^2
$
= x^3/(x^2+2x+1)

The easiest way to solve this is using long polynomial division, it leads to

= - 1/(x+1)^2 + 3/(x+1) + x - 2

you should be able to solve that.

First of all, do you know long polynomial division? If not, you could use partial fraction expansion instead.

Cheers
Rapha

3. Thanks for the quick reply. I don't know long polynomial division or partial fraction expansion (or at least not by those names). It gives me something to research though. Any extra help would be greatly appreciated.

4. Originally Posted by drew.walker

You're welcome, buddy

Originally Posted by drew.walker
I don't know long polynomial division or partial fraction expansion (or at least not by those names). It gives me something to research though. Any extra help would be greatly appreciated.
Do you know Integration by substitution? Substitute x+1 =: z, then use x = z-1 and you get

x^3/(x+1)^2 = (z-1)^3/(z)^2

You should be able to solve this integral, too. First I would calculate (z-1)^3 and then cancelling some z 's and so on....

5. Ah, that looks more familiar. I'll give it a go with that as a starting point and post my result. Let's hope it's right.

6. Wow, I think I may have actually worked it out. Is this right?

Find
$\int \frac{x^3}{(x+1)^2} dx$

Let
$z = x + 1$
$x = z - 1$

$\int \frac{(z-1)^3}{(z-1+1)^2}$
$\int \frac{(z^2 - 2z + 1)(z - 1)}{z^2}$
$\int \frac{z^3 - 3z^2 + 3z - 1}{z^2}$
$\int \frac{z^3}{z^2} - \frac{3z^2}{z^2} + \frac{3z}{z^2} - \frac{1}{z^2}$
$\int z - 3 + \frac{3}{z} - \frac{1}{z^2}$
$= \frac{z^2}{2} - 3z + 3logz + \frac{1}{z}$
$= \frac{(x+1)^2}{2} - 3(x+1) + 3log(x+1) + \frac{1}{x + 1}$

7. Yes is right! you can use this web to see if your integrator is right Wolfram Mathematica Online Integrator