Đáp án:
Giải thích các bước giải:
\(\begin{array}{l}
\int\limits_0^{\frac{\pi }{4}} {\frac{1}{{1 + \frac{{\sin x}}{{\cos x}}}}dx = } \int\limits_0^{\frac{\pi }{4}} {\frac{{\cos x}}{{\cos x + \sin x}}dx} \\
= \frac{1}{2}\mathop \smallint \nolimits_0^{\frac{\pi }{4}} (1 - \frac{{sinx - cosx}}{{sinx + cosx}})dx\\
= \frac{1}{2}\mathop \smallint \nolimits_0^{\frac{\pi }{4}} dx + \frac{1}{2}\mathop \smallint \nolimits_0^{\frac{\pi }{4}} \frac{{d(sinx + cosx)}}{{sinx + cosx}}\\
= \frac{1}{2}x|_0^{\frac{\pi }{4}} + \frac{1}{2}ln(sinx + cosx)|_0^{\frac{\pi }{4}}\\
= \frac{\pi }{8} + \frac{1}{2}ln(\sqrt 2 )
\end{array}\)
\( \to \frac{a}{b} = \frac{1}{8}:\frac{1}{2} = \frac{1}{4}\)
