Answer
$ \displaystyle \frac{1}{4}x^{4}+\frac{1}{3}x^{3}+\frac{1}{2}x^{2}+x+\ln|x-1|+C$
Work Step by Step
The numerator has degree greater than the denominator. Synthetic division:
$\begin{array}{lrrrrr}
\hline 1| & 1 & 0 & 0 & 0 & 0 \\
& & 1 & 1 & 1 & 1 \\
\hline & 1 & 1 & 1 & 1 & |\ 1 \\
\end{array}$
$\displaystyle \frac{x^{4}}{x-1}=x^{3}+x^{2}+x+1+\frac{1}{x-1}$
$\displaystyle \int\frac{x^{4}}{x-1}= \frac{1}{4}x^{4}+\frac{1}{3}x^{3}+\frac{1}{2}x^{2}+x+\ln|x-1|+C$
