$\displaystyle \int_{x=1}^{x=2} (6e^{3x}-\frac{1}{x})dx$
break it up$\displaystyle 6\int_{x=1}^{x=2} (e^{3x}dx)-\int_{x=1}^{x=2} (\frac{1}{x}dx)$
$\displaystyle 2\int (e^udu-\ln{x}+c$
is this right so far, not sure what to do after.
Well, yes, but this is a definite integral. So after you find the antiderivative $\displaystyle F(x),$ you need to apply the fundamental theorem of calculus by taking $\displaystyle F(2)-F(1).$
As for finding the antiderivative, what is $\displaystyle \int e^x\,dx?$ This is a basic integration rule, and replacing the $\displaystyle x$ with a $\displaystyle u$ won't make any difference.
No, no, your substitution was fine. I was just pointing out the integration rule:
$\displaystyle \int e^x\,dx=e^x+C.$
So,
$\displaystyle \int_1^2\left(6e^{3x}-\frac1x\right)dx$
$\displaystyle =6\int_1^2e^{3x}\,dx-\int_1^2\frac1x\,dx$
$\displaystyle =2\int_{x=1}^{x=2}e^u\,du-\bigg[\ln\lvert x\rvert\bigg]_1^2$
$\displaystyle =2\bigg[e^u\bigg]_{x=1}^{x=2}-\ln 2$
$\displaystyle =2\bigg[e^{3x}\bigg]_1^2-\ln 2$
$\displaystyle =2(e^6-e^3)-\ln 2$
$\displaystyle =2e^3(e^3-1)-\ln 2$