Đáp án đúng: A
Giải chi tiết:Đặt \(\sin x = t\) \( \Rightarrow \cos xdx = dt\).
Đổi cận: \(\left\{ \begin{array}{l}x = 0 \Rightarrow t = 0\\x = \dfrac{\pi }{2} \Rightarrow t = 1\end{array} \right.\). Khi đó ta có: \(\int\limits_0^{\frac{\pi }{2}} {\cos x} .f\left( {\sin x} \right)dx = \int_0^1 {f\left( t \right)dt} = \int\limits_0^1 {f\left( x \right)dx} \).
Theo bài ra ta có:
\(\begin{array}{l}\int\limits_0^1 {\left[ {3f\left( x \right) - 4} \right]dx} = 2\\ \Leftrightarrow 3\int\limits_0^1 {f\left( x \right)dx} - 4\int\limits_0^1 {dx} = 2\\ \Leftrightarrow 3\int\limits_0^1 {f\left( x \right)dx} - \left. {4x} \right|_0^1 = 2\\ \Leftrightarrow 3\int\limits_0^1 {f\left( x \right)dx} - 4\left( {1 - 0} \right) = 2\\ \Leftrightarrow 3\int\limits_0^1 {f\left( x \right)dx} = 6\\ \Leftrightarrow \int\limits_0^1 {f\left( x \right)dx} = 2\end{array}\)
Vậy \(\int\limits_0^{\frac{\pi }{2}} {\cos x} .f\left( {\sin x} \right)dx = \int\limits_0^1 {f\left( x \right)dx} = 2\).
Chọn A
