Cho hàm số \(f\left( x \right)\) liên tục trên R và thỏa mãn \(f\left( { - x} \right) + 2f\left( x \right) = \cos x\). Tính \(I = \int\limits_{ - \dfrac{\pi }{2}}^{\dfrac{\pi }{2}} {\cos x.f\left( x \right)dx} \)

Đặt \(t =  - x \Rightarrow dt =  - dx\), đổi cận \(\left\{ \begin{array}{l}x =  - \dfrac{\pi }{2} \Rightarrow t = \dfrac{\pi }{2}\\x = \dfrac{\pi }{2} \Rightarrow t =  - \dfrac{\pi }{2}\end{array} \right.\)

\( \Rightarrow I =  - \int\limits_{\dfrac{\pi }{2}}^{ - \dfrac{\pi }{2}} {\cos \left( { - t} \right)f\left( { - t} \right)dt}  = \int\limits_{ - \dfrac{\pi }{2}}^{\dfrac{\pi }{2}} {\cos xf\left( { - x} \right)dx} \)

\(\begin{array}{l} \Rightarrow I + 2I = \int\limits_{ - \dfrac{\pi }{2}}^{\dfrac{\pi }{2}} {\cos x.f\left( { - x} \right)dx}  + 2\int\limits_{ - \dfrac{\pi }{2}}^{\dfrac{\pi }{2}} {\cos x.f\left( x \right)dx}  = \int\limits_{ - \dfrac{\pi }{2}}^{\dfrac{\pi }{2}} {\cos x\left( {f\left( { - x} \right) + 2f\left( x \right)} \right)dx} \\ \Leftrightarrow 3I = \int\limits_{ - \dfrac{\pi }{2}}^{\dfrac{\pi }{2}} {{{\cos }^2}xdx}  = \int\limits_{ - \dfrac{\pi }{2}}^{\dfrac{\pi }{2}} {\dfrac{{1 + \cos 2x}}{2}dx}  = \left. {\left( {\dfrac{1}{2}x + \dfrac{{\sin 2x}}{4}} \right)} \right|_{ - \dfrac{\pi }{2}}^{\dfrac{\pi }{2}} = \dfrac{\pi }{4} + \dfrac{\pi }{4} = \dfrac{\pi }{2}\\ \Rightarrow I = \dfrac{\pi }{6}\end{array}\)