I have a problem with solving this integral by partial fraction I have an example like this :
$\int \frac{1}{x^{4}}dx$ = $\int x^{-4}dx$ = $\frac{1}{-4 + 1}x^{-4 + 1}+C$ = $\frac{1}{-3}x^{-3}+C$
What if this?
$\int \frac{1}{2x-10}dx$ = ?

$\displaystyle \int\! x^r \, dx=\frac{x^{r+1}}{r+1}+C$ for $\color{red}{r \ne -1}$. 
Let $u = 2x-10$ then $dx = \frac{1}{2}du$ and 
$\displaystyle \int \frac{1}{2x-10}dx=\frac{1}{2}\int \frac{1}{u}\,du =\frac{1}{2}\ln|u|+C = \frac{1}{2}\ln|2x-10|+C.   $
