Tích phân \(I = \int\limits_{\frac{\pi }{3}}^{\fra...

C

- Hướng dẫn giải

Xét \({I_1} = \int\limits_{\frac{\pi }{3}}^{\frac{\pi }{2}} {\frac{{\cos x}}{{\sin x + \cos x}}dx} \)

Ta có:

\(\begin{array}{l}\left\{ \begin{array}{l}{I_2} = I + {I_1} = \int\limits_{\frac{\pi }{3}}^{\frac{\pi }{2}} {dx} \\{I_3} = {I_1} - I = \int\limits_{\frac{1}{2} + \frac{{\sqrt 3 }}{2}}^1 {\frac{1}{t}dt} \end{array} \right.\\ \Rightarrow I = \frac{{{I_2} - {I_3}}}{2} = \frac{\pi }{{12}} - \frac{{\ln \frac{{1 + \sqrt 3 }}{2}}}{2},{\rm{ }}t = \sin x + \cos x\end{array}\)