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How does one integrate [math]\displaystyle \int\dfrac{1}{\sqrt{x^2-1}}\,dx?[/math]
How does that lead to the final answer?
After substituting [math]u = \sec \theta + \tan \theta[/math], you get [math]\int \frac{du}{u} = \ln |u| + C = \ln |\sec \theta + \tan \theta| + C[/math].