Evalute : e^x [( x)^2+2 x x ] d x - Vidyadip

Question

Evalute : $\int e^x\left[(\log x)^2+\frac{2 \log x}{x}\right] d x$

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Answer

$
\text { Let } I=\int e^x\left[(\log x)^2+\frac{2 \log x}{x}\right] d x
$
Put $f(x)=(\log x)^2$
Then $f^{\prime}(x)=\frac{d}{d x}(\log x)^2=2 \log x \cdot \frac{d}{d x}(\log x)$
$
\begin{aligned}
& =2 \log x \times \frac{1}{x}=\frac{2 \log x}{x} \\
\therefore I & =\int e^x\left[f(x)+f^{\prime}(x)\right] d x \\
& = e ^x \cdot f(x)+c=e^x \cdot(\log x)^2+c
\end{aligned}
$

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