Some Questions on Calculus

\hspace{-16}\bf{(1)\;\; \int_{e^{e^{e}}}^{\infty}\frac{1}{x.(\log x).({\log \log x}).(\log \log \log x)^{\frac{4}{3}}}dx}$\\\\\\ $\bf{(2)}$ For Which Integer $\bf{ 1\leq m\leq 10}$ is it true that\\\\\\ $\bf{\int_{0}^{\pi}(\cos x).(\cos 2x).(\cos 3x)......(\cos mx)dx = 0}$\\\\\\ $\bf{(3)}$ Let $\bf{f(x)=\sum_{n=0}^{\infty}\frac{\sin (nx)}{n!}.\;}$ Then $\bf{f\left(\frac{\pi}{3}\right)=}$