Let \(f(x) = \frac{\cos x}{1 + \sin x}\).

(a) Show that \(f'(x) < 0\) for all \(x\) in the interval \(-\frac{1}{2}\pi < x < \frac{3}{2}\pi\).

(b) Find \(\int_{\frac{1}{6}\pi}^{\frac{1}{2}\pi} f(x) \, dx\). Give your answer in a simplified exact form.