Answer
$\frac{\pi}{3}$
Work Step by Step
Since $u=4x$, then $du= 4dx$, and hence when $x:1/4 \to 1/2$ then $u:1\to 2$. Now, we have
$$\int_{1/4}^{1/2}\frac{ dx}{x\sqrt{16x^2-1}}=\frac{1}{4}\int \frac{ du}{u\sqrt{u^2-1}}\\=\sec^{-1}u|_1^2
=\sec^{-1}2-\sec^{-1}1=\frac{\pi}{3}-0=\frac{\pi}{3}.$$
