Câu hỏi 13

Ta có : \(f\left( x \right) = \int {f'\left( x \right)dx}  = \int {\dfrac{{\ln x}}{x}dx} \)

Đặt \(t = \ln x \Rightarrow dt = \dfrac{{dx}}{x}\) \( \Rightarrow \int {\dfrac{{\ln x}}{x}dx}  = \int {tdt}  = \dfrac{{{t^2}}}{2} + C = \dfrac{{{{\ln }^2}x}}{2} + C\)

\( \Rightarrow f\left( x \right) = \dfrac{{{{\ln }^2}x}}{2} + C\). Mà \(f\left( 1 \right) = \dfrac{3}{2} \Rightarrow C = \dfrac{3}{2} \Rightarrow f\left( x \right) = \dfrac{{{{\ln }^2}x}}{2} + \dfrac{3}{2}\)

Vậy \(f\left( 3 \right) = \dfrac{{{{\ln }^2}3}}{2} + \dfrac{3}{2} = \dfrac{{{{\ln }^2}3 + 3}}{2}\) .

Chọn D.