Evalute : ( x)^2 d x - Vidyadip

Question

Evalute : $\int(\log x)^2 d x$

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Answer

$
\begin{aligned}
& \int(\log x)^2 d x=\int(\log x)^2 \cdot 1 d x \\
= & (\log x)^2 \int 1 d x-\int\left[\frac{d}{d x}(\log x)^2 \cdot \int 1 d x\right] d x \\
= & (\log x)^2 \cdot x-\int\left[2 \log x \cdot \frac{d}{d x}(\log x) \times x\right] d x \\
= & x(\log x)^2-\int 2 \log x \times \frac{1}{x} \times x d x \\
= & x(\log x)^2-2 \int(\log x) \cdot 1 d x \\
= & x(\log x)^2-2\left\{(\log x) \int 1 d x-\left[\frac{d}{d x}(\log x) \int 1 d x\right] d x\right\} \\
= & x(\log x)^2-2\left\{(\log x) \cdot x-\int \frac{1}{x} \times x d x\right\} \\
= & x(\log x)^2-2 x \log x+2 \int 1 d x \\
= & x(\log x)^2-2 x \log x+2 x+c .
\end{aligned}
$

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