CÂU HỎI: Một nguyên hàm F(x) của hàm số f(x) = a + bcos2x thỏa mãn \(F\left( 0 \right) = \frac{{\rm{\pi }}}{2},\;F\left( {\frac{{\rm{\pi }}}{2}} \right) = \frac{{\rm{\pi }}}{6},\;F\left( {\frac{{\rm{\pi }}}{{12}}} \right) = \frac{{\rm{\pi }}}{3}\) là

Ta có 

\(F\left( x \right){\rm{ = a}}x + \frac{b}{2}\sin 2x + C,\left\{ \begin{array}{l}
F\left( 0 \right) = \frac{\pi }{2}\\
F\left( {\frac{\pi }{2}} \right) = \frac{\pi }{6}\\
F\left( {\frac{\pi }{{12}}} \right) = \frac{\pi }{3}
\end{array} \right. \Leftrightarrow \left\{ \begin{array}{l}
a =  – \frac{2}{3}\\
b = \frac{{7\pi }}{9}\\
c = \frac{\pi }{2}
\end{array} \right.\)

Vậy \(F\left( x \right) =  – \frac{2}{3}x + \frac{{7{\rm{\pi }}}}{9}\sin 2x + \frac{{\rm{\pi }}}{2}\)