In the integral $\displaystyle\int_0^{\pi/2} x\cot x\,dx$

$$\displaystyle\int_0^{\pi/2} x\cot x\,dx$$

By integration by parts I get a part $[x \ln|\sin x|]_0^{\pi/2}$ where $(\pi/2)\ln|\sin(\pi/2)|=0$ but, does $0\cdot \ln|\sin0|=0$ or is it indeterminate form of $0\cdot\infty$? Is there any way to find the above integral other than the integration by parts? I prefer if it is in elementary functions as I'm still a new student to Integration.

Your integral is an improper integral because $\cot x$ is not continuous at $x=0$. To correct this issue, you should set up the problem as $$\int_0^{\pi/2}x\cot x\,dx=\lim_{a\to 0^+}\int_a^{\pi/2}x\cot x\,dx$$ Then evaluate the integral $$\int_a^{\pi/2}x\cot x\,dx$$ This will give you an expression in $a$. Once you have this expression take $a\to 0$.
The function you have to integrate is not defined at $0$, but it is bounded on $(0,\frac{\pi}{2}]$, so this is a regular Riemann Integral, not an improper one. For $0<h<\frac{\pi}{2}$ the function is continuous on $[h,\frac{\pi}{2}]$, so you can apply the Fundamental Theorem of Calculus, and then take the limit for $h\rightarrow 0^+$ of the function you get.