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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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$\int x.\log xdx= \log x.\int xdx - \int(\frac{d(\log x)}{dx}.\int x dx)dx\\ \\ \int x\log xdx = \log x.\frac{x^2}{2}- \int (\frac{1}{x}.\frac{x^2}{2})dx\\ \int x\log xdx = \log x.\frac{x^2}{2}- \int \frac{x}{2}dx\\ \int x\log xdx = \log x.\frac{x^2}{2}- \frac{x^2}{4}+ C$

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