Lời giải

+ \( x \in \left[\frac{\pi}{6}, \frac{\pi}{3}\right]: \quad \frac{1}{\cos^2 x} = \frac{1}{\sin^2 x} \iff \sin x = \cos x \iff x = \frac{\pi}{4}\)

\(S = \int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \left| \frac{1}{\cos^2 x} - \frac{1}{\sin^2 x} \right| dx = \int_{\frac{\pi}{6}}^{\frac{\pi}{4}} \left| \frac{1}{\cos^2 x} - \frac{1}{\sin^2 x} \right| dx + \int_{\frac{\pi}{4}}^{\frac{\pi}{3}} \left| \frac{1}{\cos^2 x} - \frac{1}{\sin^2 x} \right| dx\)

    \(= \left| \int_{\frac{\pi}{6}}^{\frac{\pi}{4}} \left( \frac{1}{\cos^2 x} - \frac{1}{\sin^2 x} \right) dx \right| + \left| \int_{\frac{\pi}{4}}^{\frac{\pi}{3}} \left( \frac{1}{\cos^2 x} - \frac{1}{\sin^2 x} \right) dx \right| \)

    \(= \Bigg| \left( \tan(x) + \cot(x) \right) \Big|_{\frac{\pi}{6}}^{\frac{\pi}{4}} \Big| + \Big| \left( \tan(x) + \cot(x) \right) \Big|_{\frac{\pi}{4}}^{\frac{\pi}{3}} \Bigg| \)

    \(= \left| 2 - \left(\frac{\sqrt{3}}{3} + \sqrt{3}\right) \right| 
+ \left| \left(\sqrt{3} + \frac{\sqrt{3}}{3}\right) - 2 \right|\)

    \(= 2 .\left( \frac{4\sqrt{3}}{3} - 2 \right) \) (đvdt)

* Bấm:  \( \int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \left| \frac{1}{\cos^2 x} - \frac{1}{\sin^2 x} \right| dx = \text{(dùng máy tính, 35 giây)} \quad 0.618802\)

\(= \frac{8\sqrt{3}}{3} - 4 \approx 0.618802\)