Evaluate: log 2x dx. - Vidyadip

Question

Evaluate:
$\int\text{x log 2x dx}$.

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Answer

$\text{I}=\int\log2\text{x}\cdot\text{x dx}=\log2\text{x}\cdot\frac{\text{x}^{2}}{2}-\int\frac{1}{\text{x}}\cdot\frac{\text{x}^{2}}{2}\text{dx}+\text{c}_{1}$
= $\frac{\text{x}^{2}}{2}\cdot\log\text{2x}-\frac{1}{2}\int\text{x dx + c}_{1}=\frac{\text{x}^{2}}{2}\cdot\log\text{ 2x}-\frac{\text{x}^{2}}{4}+\text{c}.$

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