Question

\[\int\limits \frac{dx}{x \sqrt{x ^{2}-16}}\]

Answer 1

\(x = 4\sec{\theta} \implies \text{d}x = 4\sec{\theta}\tan{\theta}\text{d}\theta\)

Answer 2

You are looking for a substitution that will help you to get rid of that radical, let me show you why @FoolAroundMath substitution works.

If \(x=4\sec(\theta) \) then the radical becomes
\[ 4\sqrt{\sec^2(\theta) -1}=4 \tan(\theta)\]