Evalute : d x x [( x)^2+4 x-1 ] - Vidyadip

Question

Evalute : $\int \frac{d x}{x\left[(\log x)^2+4 \log x-1\right]}$

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Answer

Let $I=\int \frac{d x}{x\left[(\log x)^2+4 \log x-1\right]}$
$
=\int \frac{1}{(\log x)^2+4 \log x-1} \cdot \frac{1}{x} d x
$
Put $\log x=t \quad \therefore \frac{1}{x} d x=d t$
$
\begin{aligned}
\therefore I & =\int \frac{1}{t^2+4 t-1} d t \\
& =\int \frac{1}{\left(t^2+4 t+4\right)-5} d t \\
& =\int \frac{1}{(t+2)^2-(\sqrt{5})^2} d t \\
& =\frac{1}{2 \sqrt{5}} \log \left|\frac{t+2-\sqrt{5}}{t+2+\sqrt{5}}\right|+c
\end{aligned}
$
$
=\frac{1}{2 \sqrt{5}} \log \left|\frac{\log x+2-\sqrt{5}}{\log x+2+\sqrt{5}}\right|+c
$

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