Evaluate the following integrals:

Question:

Evaluate the following integrals:

$\int \frac{\mathrm{e}^{\mathrm{x}}}{\mathrm{x}}\left\{\mathrm{x}(\log \mathrm{x})^{2}+2 \log \mathrm{x}\right\} \mathrm{dx}$

Solution:

Let $I=\int \frac{e^{x}}{x}\left\{x(\log x)^{2}+2 \log x\right\} d x$

$=\int e^{x}(\log x)^{2} d x+2 \int \frac{e^{x}}{x} \log x d x$

Using integration by parts,

$=e^{x}(\log x)^{2}-\int e^{x} \frac{d}{d x}(\log x)^{2}+2 \int \frac{e^{x}}{x} \log x d x$

$=e^{x}(\log x)^{2}-2 \int \frac{e^{x}}{x} \log x d x+2 \int \frac{e^{x}}{x} \log x d x$

$=e^{x}(\log x)^{2}+c$

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