If ∫√(sec 2x - 1) dx = α log_e |cos 2x + β + √(cos 2x (1 + cos 2x / β))| + constant, then β - α is equal to ____.
Step-by-Step Solution
Key Concept: Use the identity sec 2x - 1 = (1 - cos 2x) / cos 2x = 2 sin^2 x / cos 2x. Then the integral becomes \int(\sqrt{2} sin x) / \sqrt{cos 2x} dx. Substitute cos 2x = t.
The integral is \int\sqrt{sec 2x - 1} dx = \int\sqrt{ (1-cos 2x}/cos 2x ) dx = \int\sqrt{2 sin^2 x / cos 2x} dx = \sqrt{2} \int sin x / \sqrt{1 - 2 sin^2 x} dx. Let cos x = u, then -sin x dx = du. Alternatively, use the substitution cos 2x = t, then -2 sin 2x dx = dt. The integral simplifies to a standard form leading to the logarithmic expression. Comparing the coefficients with the given form, we find \alpha and \beta.
Correct Answer: 1