Cho hàm số $f(x)$ có đạo hàm liên tục trên \(\le...

D

- Hướng dẫn giải

Xét $\int\limits_0^{\frac{\pi }{2}} {f'\left( x \right){{\cos }^2}x{\rm{d}}x}  = 10$, đặt $\left\{ \begin{array}{l} u = {\cos ^2}x\\ {\rm{d}}v = f'\left( x \right){\cos ^2}x{\rm{d}}x \end{array} \right. \Rightarrow \left\{ \begin{array}{l} {\rm{d}}u =  - \sin 2x{\rm{d}}x\\ v = f\left( x \right) \end{array} \right..$

Khi đó $10 = \int\limits_0^{\frac{\pi }{2}} {f'\left( x \right){{\cos }^2}x{\rm{d}}x}  = {\cos ^2}xf\left( x \right)\left| {\begin{array}{*{20}{c}} {\frac{\pi }{2}}\\ 0 \end{array}} \right. + \int\limits_0^{\frac{\pi }{2}} {f\left( x \right)\sin 2x{\rm{d}}x} {\rm{ }}$

$\Leftrightarrow 10 =  - f\left( 0 \right) + \int\limits_0^{\frac{\pi }{2}} {f\left( x \right)\sin 2x{\rm{d}}x}  \to \int\limits_0^{\frac{\pi }{2}} {f\left( x \right)\sin 2x{\rm{d}}x}  = 10 + f\left( 0 \right) = 13.$