Question
Evaluate the following integrals:
$\int\frac{\text{e}^{\text{x}}}{\text{x}}\Big\{\text{x}(\log\text{x})^2+2\log\text{x}\Big\}\text{dx}$
$\int\frac{\text{e}^{\text{x}}}{\text{x}}\Big\{\text{x}(\log\text{x})^2+2\log\text{x}\Big\}\text{dx}$
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Answer
We have,
$\text{I}=\int\frac{\text{e}^{\text{x}}}{\text{x}}\Big\{\text{x}(\log\text{x})^2+2\log\text{x}\Big\}\text{dx}$
$=\int\text{e}^{\text{x}}\Big\{(\log\text{x})^2+\frac{2}{\text{x}}\log\text{x}\Big\}\text{dx}$
$=\int\text{e}^{\text{x}}(\log\text{x})^2+2\int\frac{\text{e}^{\text{x}}}{\text{x}}\log\text{x dx}$
Integrating by parts
$=\text{e}^{\text{x}}(\log\text{x})^2-\int\text{e}^{\text{x}}\frac{\text{d}}{\text{dx}}(\log\text{x})^2\text{dx}+2\int\text{e}^{\text{x}}\frac{1}{\text{x}}\log\text{x dx}$
$=\text{e}^{\text{x}}(\log\text{x})^2-\int\text{e}^{\text{x}}\frac{2\log\text{x}}{\text{x}}\text{dx}+2\int\text{e}^\text{x}\frac{\log\text{x}}{\text{x}}\text{dx}$
$=\text{e}^{\text{x}}(\log\text{x)}^2+\text{C}$
$\text{I}=\int\frac{\text{e}^{\text{x}}}{\text{x}}\Big\{\text{x}(\log\text{x})^2+2\log\text{x}\Big\}\text{dx}$
$=\int\text{e}^{\text{x}}\Big\{(\log\text{x})^2+\frac{2}{\text{x}}\log\text{x}\Big\}\text{dx}$
$=\int\text{e}^{\text{x}}(\log\text{x})^2+2\int\frac{\text{e}^{\text{x}}}{\text{x}}\log\text{x dx}$
Integrating by parts
$=\text{e}^{\text{x}}(\log\text{x})^2-\int\text{e}^{\text{x}}\frac{\text{d}}{\text{dx}}(\log\text{x})^2\text{dx}+2\int\text{e}^{\text{x}}\frac{1}{\text{x}}\log\text{x dx}$
$=\text{e}^{\text{x}}(\log\text{x})^2-\int\text{e}^{\text{x}}\frac{2\log\text{x}}{\text{x}}\text{dx}+2\int\text{e}^\text{x}\frac{\log\text{x}}{\text{x}}\text{dx}$
$=\text{e}^{\text{x}}(\log\text{x)}^2+\text{C}$
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