Tính P = (a − 2 b)/b .

9/10

Biết \(\int\limits_{\frac{\pi }{4}}^{\frac{\pi }{3}} {\frac{1}{{{{\sin }^2}x.{{\cos }^2}x}}dx = \frac{{a\sqrt 3 }}{b}} \) (a, b ℤ). Tính \(P = \frac{{a - 2b}}{b}\).

\(P = \frac{4}{3}\);

\(P = - \frac{4}{3}\);

\(P = - \frac{2}{3}\);

\(P = \frac{2}{3}\).

Giải thích

Đáp án đúng là: B

\(\int\limits_{\frac{\pi }{4}}^{\frac{\pi }{3}} {\frac{1}{{{{\sin }^2}x.{{\cos }^2}x}}dx = } \int\limits_{\frac{\pi }{4}}^{\frac{\pi }{3}} {\frac{{{{\sin }^2}x + {{\cos }^2}x}}{{{{\sin }^2}x.{{\cos }^2}x}}dx = } \)\(\int\limits_{\frac{\pi }{4}}^{\frac{\pi }{3}} {\left( {\frac{1}{{{{\cos }^2}x}} + \frac{1}{{{{\sin }^2}x}}} \right)dx} \)\( = \left. {\left( {\tan x - \cot x} \right)} \right|_{\frac{\pi }{4}}^{\frac{\pi }{3}} = \frac{{2\sqrt 3 }}{3}\).

Do đó \(P = \frac{{2 - 2.3}}{3} = - \frac{4}{3}\).