Phương pháp giải:

Sử dụng phương pháp tích phân 2 vế.

Lời giải chi tiết:

Ta có: \(3xf\left( {{x^2}} \right) - f\left( x \right) = 9{x^3} - 1\), (\(f\left( x \right)\) liên tục trên \(\left[ {0;1} \right]\))

\( \Rightarrow \int\limits_0^1 {\left( {3xf\left( {{x^2}} \right) - f\left( x \right)} \right)} dx = \int\limits_0^1 {\left( {9{x^3} - 1} \right)} dx \Leftrightarrow 3\int\limits_0^1 {xf\left( {{x^2}} \right)} dx - \int\limits_0^1 {f\left( x \right)} dx = \left. {\left( {\dfrac{9}{4}{x^4} - x} \right)} \right|_0^1\)

\( \Leftrightarrow \dfrac{3}{2}\int\limits_0^1 {f\left( {{x^2}} \right)} d\left( {{x^2}} \right) - \int\limits_0^1 {f\left( x \right)} dx = \dfrac{5}{4} \Leftrightarrow \dfrac{3}{2}\int\limits_0^1 {f\left( t \right)} dt - \int\limits_0^1 {f\left( x \right)} dx = \dfrac{5}{4},\,\,\left( {t = {x^2}} \right)\)

\( \Leftrightarrow \dfrac{3}{2}\int\limits_0^1 {f\left( x \right)} dx - \int\limits_0^1 {f\left( x \right)} dx = \dfrac{5}{4} \Leftrightarrow \dfrac{1}{2}\int\limits_0^1 {f\left( x \right)} dx = \dfrac{5}{4} \Leftrightarrow \int\limits_0^1 {f\left( x \right)} dx = \dfrac{5}{2}\).

Chọn: A