Find the value of _a x d x. - Vidyadip

Question

Find the value of $\int \log _a x d x$.

✓

Answer

Find the value of $\int \log _a x d x$. Let $\quad I =\int \log _a x d x$
On changing the base
$
I=\int \log _e x \cdot \log _a e d x
$
$
=\log _a e \int_{II} 1 \cdot \log _e x d x
$
Taking 1 as the second function, integrating by parts :
$
\begin{aligned}
I & =\log _a e\left[\log _e x \cdot x-\int \frac{1}{x} \cdot x d x\right] \\
& =\log _a e\left(x \log _e x-x\right)+C \\
& =x \log _a e\left(\log _e x-1\right)+C \text {}
\end{aligned}
$

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