\(\displaystyle \int 1+ \frac{6}{6-x} dx\)

Well you are right that integration can be separated when two expressions are added or subtracted, integration has this additive property. The equation would become

\(\displaystyle \int 1 dx + \int \frac{6}{6-x} dx\)

But you made a little mistake with your fraction, the fraction can be changed to

\(\displaystyle \int 1 dx + \int \frac{3}{\frac{1}{2}(6-x)} dx\)

Changing the fraction doesn't really help with solving it though, try the change of variable y=6-x