Cho \(\int\limits_0^{\frac{\pi }{6}} {{{\sin }^n}x.\cos xdx}  = \frac{1}{{64}}\). Tìm n?

* Hướng dẫn giải

\(\begin{array}{l}
\frac{1}{{64}} = \int\limits_0^{\frac{\pi }{6}} {{{\sin }^n}x.\cos xdx}  = \int\limits_0^{\frac{\pi }{6}} {{{\sin }^n}xd\left( {\sin x} \right)} \\
 = \frac{{{{\sin }^{n + 1}}x}}{{n + 1}}\left| {_0^{\frac{\pi }{6}}} \right. = \frac{1}{{n + 1}}.{\left( {\frac{1}{2}} \right)^{n + 1}}\\
 \Rightarrow n = 3
\end{array}\)