Question

$\int \frac{1}{x\sqrt{4-9{\left(\log x\right)}^2}}dx$

Answer 1

$$\begin{gathered} \int \frac{dx}{x\sqrt{4-9{\left(\log x\right)}^2}} \\ \mathrm{let}\log x=t \\ \Rightarrow \frac{1}{x}dx=dt \\ \mathrm{Now},\int \frac{dx}{x\sqrt{4-9{\left(\log x\right)}^2}} \\ =\int \frac{dt}{\sqrt{4-9t^2}} \\ =\int \frac{dt}{\sqrt{2^2-{\left(3t\right)}^2}} \\ =\frac{1}{3}{\sin }^{-1}\left(\frac{3t}{2}\right)+C \\ =\frac{1}{3}{\sin }^{-1}\left(\frac{3\log x}{2}\right)+C \end{gathered}$$