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$\\\int \left(\log \left(\log \left(x\right)\right)+\frac{1}{\left(\log \left(x\right)\right)^2}\right)dx\\put\:\log \left(x\right)=t\\x=e^t\Rightarrow \:\:\\dx=e^tdt\\\Rightarrow \int \:e^t\left(\log \left(t\right)+\frac{1}{^{t^2}}\right)dt\:\\\Rightarrow \int \:\:e^t\left(\log \:\left(t\right)+\frac{1}{t}-\frac{1}{t}+\frac{1}{^{t^2}}\right)dt\\\Rightarrow \int \:e^t\left(\log \left(t\right)+\frac{1}{t}\right)dt-\int \:e^t\left(\frac{1}{t}+\left(-\frac{1}{t^2}\right)\right)dt\:\:\:\:\\now\:apply\\\int \:e^x\left(f\left(x\right)+f'\left(x\right)\right)dx=e^xf\left(x\right)+c$

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$\begin{array}{lll}\lim_{x \rightarrow 0} x*\log(x) & = & \lim_{x\rightarrow 0} \frac{\log(x)}{x^{-1}} \\ { } & = & \lim_{x \rightarrow 0} \frac{x^{-1}}{-x^{-2}} \\ {} & = & \lim_{x\rightarrow 0}-x \\ { } & = & 0\end{array}$

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