Question

integral dx/x^2 sqrt (x^2 - 2)

Answer 1

Assuming you mean
\[\int\frac{dx}{x^2\sqrt{x^2-2}}\]
Try letting \(x=\sqrt 2\sec u\), so that \(dx=\sqrt2\sec u\tan u~du\).
\[\int\frac{\sqrt2\sec u\tan u}{2\sec^2u\sqrt{2\sec^2u-2}}~du\\
\frac{1}{2}\int\frac{\sec u\tan u}{\sec^2u\sqrt{\sec^2u-1}}~du\\
\frac{1}{2}\int\frac{\sec u\tan u}{\sec^2u\sqrt{\tan^2u}}~du\\
\frac{1}{2}\int\frac{\sec u}{\sec^2u}~du\\
\frac{1}{2}\int\cos u~du
\]