Câu hỏi 2

Ta có: \(F\left( x \right) = \int {f\left( x \right)dx}  = \int {\frac{{dx}}{{x\ln x}}} \).

Đặt \(\ln x = t \Rightarrow \frac{{dx}}{x} = dt\)

\( \Rightarrow \int {\frac{{dx}}{{x\ln x}}}  = \int {\frac{{dt}}{t}}  = \ln \left| t \right| + C = \ln \left| {\ln x} \right| + C = \left\{ \begin{array}{l}\ln \left( {\ln x} \right) + {C_1}\,\,khi\,\,x > 1\\\ln \left( { - \ln x} \right) + {C_2}\,\,khi\,\,x < 1\end{array} \right.\)

+) \(F\left( {\frac{1}{e}} \right) = 2 \Leftrightarrow \ln \left( { - \ln \frac{1}{e}} \right) + {C_1} = 2 \Leftrightarrow {C_1} = 2\).

+) \(F\left( e \right) = \ln 2 \Leftrightarrow \ln \left( {\ln e} \right) + {C_2} = \ln 2 \Leftrightarrow {C_2} = \ln 2\)

\( \Rightarrow F\left( x \right) = \left\{ \begin{array}{l}\ln \left( {\ln x} \right) + 2\,\,\,\,\,\,\,\,\,\,khi\,\,x > 1\\\ln \left( { - \ln x} \right) + \ln 2\,\,khi\,\,x < 1\end{array} \right.\)

\( \Rightarrow F\left( {\frac{1}{{{e^2}}}} \right) = \ln \left( { - \ln \frac{1}{{{e^2}}}} \right) + \ln 2 = 2\ln 2\) và \(F\left( {{e^2}} \right) = \ln \left( {\ln {e^2}} \right) + 2 = \ln 2 + 2\)

\( \Rightarrow F\left( {\frac{1}{{{e^2}}}} \right) + F\left( {{e^2}} \right) = 2\ln 2 + \ln 2 + 2 = 3\ln 2 + 2\).

Chọn A.