Question: * Given a triangle $\mathrm{A}\left(2,0\right),\ \mathrm{B}\left(1,3\right),\
\mathrm{C}\left(5,2\right)\ \mbox{with}\ \rho\left(x,y\right) = x$; I need to find it's centre of mass ?.
* I know I need to integrate the density formula over the region, but I don't understand how to get the limits for the integrals to calculate the area.
* Do I need to find formulas for the lines and somehow use those ?.

Answer: 
Denote by $\;AB,AC,BC\;$ the respective lines on which the sides $\;AB,AC,BC\;$ lie, thus:

$$\begin{cases}AB:\;y=-3x+6\\{}\\AC:\;y=\cfrac23x-\cfrac43\\{}\\BC:\;y=-\cfrac14x+\cfrac{13}4\end{cases}$$

You should try to do a diagram, and then you need for the mass you need the integrals

$$M=\int\_1^2\int\_{-3x+6}^{-\frac14x+\frac{13}4} x\,dy\,dx+\int\_2^5\int\_{\frac23x-\frac43}^{-\frac14x+\frac{13}4}x\,dy\,dx$$

Take it from here.
Yes find lines equation is a way to set the correct integral. 

Firstly make a graph of the points and then find the equations you need.