Evaluate _1^3 x^2 x d x - Vidyadip

Question

Evaluate $\int_1^3 x^2 \cdot \log x \ d x$

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Answer

$\text { Let } I =\int_1^3 x^2 \cdot \log x \ d x$
$=\left[\log x \int x^2 d x\right]_1^3-\int_1^3\left[\frac{ d }{ d x}(\log x) \int x^2 d x\right] d x$
$=\left[\log x \cdot \frac{x^3}{3}\right]_1^3-\int_1^3 \frac{1}{x} \cdot \frac{x^3}{3} d x$
$=\left[9 \log 3-\log 1 \cdot \frac{1}{3}\right]-\frac{1}{3} \int_1^3 x^2 d x$
$=(9 \log 3-0)-\frac{1}{3}\left[\frac{x^3}{3}\right]_1^3$
$=9 \log 3-\frac{1}{3}\left(\frac{27}{3}-\frac{1}{3}\right)$
$=9 \log 3-\frac{1}{3}\left(\frac{26}{3}\right)$
$\therefore I =9 \log 3-\frac{26}{9}$

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