Evalute : d x 25 x-x( x)^2 - Vidyadip

Question

Evalute : $\int \frac{d x}{25 x-x(\log x)^2}$

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Answer

$
\text { Let } \begin{aligned}
I & =\int \frac{d x}{25 x-x(\log x)^2} \\
& =\int \frac{1}{25-(\log x)^2} \cdot \frac{1}{x} d x
\end{aligned}
$
Put $\log x=t \quad \therefore \frac{1}{x} d x=d t$
$
\begin{aligned}
\therefore I & =\int \frac{1}{25-t^2} d t \\
& =\frac{1}{2 \times 5} \log \left|\frac{5+t}{5-t}\right|+c \\
& =\frac{1}{10} \log \left|\frac{5+\log x}{5-\log x}\right|+c .
\end{aligned}
$

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