# Thread: Simple Integral question w/ parts

1. Originally Posted by Chris L T521

Where are you stuck in the division process?

--Chris
Well, I guess the whole thing, like how do you get $\displaystyle 1-\frac {1}{1+s^2}$
It's something to do with the s^2 right?

2. Originally Posted by matt3D
Well, I guess the whole thing, like how do you get $\displaystyle 1-\frac {1}{1+s^2}$
It's something to do with the s^2 right?
You take $\displaystyle 1+s^2~|\overline{~~~~s^2}$

We see that $\displaystyle s^2$ goes into $\displaystyle s^2$ 1 time.

Thus, we should get

$\displaystyle .~~~~~~~~~~~~~~~~~~{\color{blue}1}$
$\displaystyle {\color{magenta}1+s^2}~|\overline{~~~~~~~~~s^2}$
$\displaystyle .~~~~~~~~~\underline{-(1+s^2)}$
$\displaystyle .~~~~~~~~~~~~~~~~~{\color{red}-1}$

So our quotient will be $\displaystyle {\color{blue}1}+\frac{{\color{red}-1}}{{\color{magenta}1+s^2}}=1-\frac{1}{1+s^2}$, where $\displaystyle -\frac{1}{1+s^2}$ is the remainder.

Does this make sense?

--Chris

3. This is done by polynomial long division. Check the link here to see how it is done: Polynomial Long Division

4. If you want to do it quickly without polynomial division:

$\displaystyle \frac{x^2}{x^2+1}$

Add one/subtract one: $\displaystyle \frac{(x^2 + 1) - 1}{x^2+1}$
Split: $\displaystyle \frac{x^2+1}{x^2+1} - \frac{1}{x^2+1}$
Cancel: $\displaystyle 1 - \frac{1}{x^2+1}$

5. Originally Posted by Chop Suey
If you want to do it quickly without polynomial division:

$\displaystyle \frac{x^2}{x^2+1}$

Add one/subtract one: $\displaystyle \frac{(x^2 + 1) - 1}{x^2+1}$
Split: $\displaystyle \frac{x^2+1}{x^2+1} - \frac{1}{x^2+1}$
Cancel: $\displaystyle 1 - \frac{1}{x^2+1}$
Aha! I remember that trick now...I remember seeing it before...

--Chris

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