Evaluate _1^3 x d x - Vidyadip

Question

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

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Answer

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

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