Polynomial Long Division Confusion (simplifying $\frac{x^{5}}{x^{2}+1}$) I need to simplify \begin{equation}
\frac{x^{5}}{x^{2}+1}
\end{equation}
by long division in order to solve an integral. 
However, I keep getting an infinite series:
\begin{equation}
x^{3}+x+\frac{1}{x}-\frac{1}{x^{3}}+...
\end{equation}

Why wouldn't you get an infinite series?  
If $x^2 + 1$ doesn't divide evenly into $x^5$ (which it doesn't) you will get a remainder.  Just like with numbers with remainders if you try to continue you will get a decimal.  Here if try to divide into the remainder you will get an expression with a negative power.
The thing is when you get a remainder that can be divided into any further.... you stop.  And you let it be simply a remainder.
$\frac {x^5}{x^2+ 1} = \frac {x^5 + x^3}{x^2 + 1} -\frac {x^3}{x^2 + 1}$
$= x^3 - \frac {x^3 + x}{x^2 + 1} + \frac {x}{x^2+ 1} =$
$x^3 - x + \frac {x}{x^2 + 1}$.
Now we can't divide any further as the degree of the denominator ($x^2 + 1$) is $2$ and that is larger than the degree of that numerator ($x$).  So we are done.
$\frac {x^5}{x^2 +1} = x^3 - x +\frac {x}{x^2 +1}$.
Put another way:  $x^5 = (x^3 - x)(x^2 + 1) + x$.
$x$ is .... just a remainder you cant do any thing with.
It is exactly like.
$\frac {249}{7} = \frac {210 + 39}{7} = \frac {210}7 + \frac {39}7=$
$30 + \frac {35 + 4}{7} = 30 + \frac {35}7 + \frac 47=$
$30 + 5 + \frac 47 = 35\frac 47$.  
We've divided as far as we can go. 
If you tried to go further we would get decimals:
$30 + 5 + \frac {40}{7*10} = 30 + 5 + {35 + 5}{70} = $
$30 + 5 + \frac 5{10} + \frac 5{70} =30 + 5 + \frac 5{10} + \frac {50}{700} =$
$30 + 5 + \frac 5{10} + \frac 7{100} + \frac 1{1000} + ......$.
$= 35.571428571428571428571428571429.....$
But we weren't asked to go that for and as we aren't masochists.... we stopped at $35\frac 47$.
$
