Cho hàm số f(x) xác định trên $\Large\mathrm{ (0 ;+\infty) \backslash\

Cho hàm số f(x) xác định trên $\Large\mathrm{ (0 ;+\infty) \backslash\{e\},}$ thỏa mãn $\Large\mathrm{f^{\prime}(x)=\dfrac{1}{x(\ln x-1)} ,f\left(e^{2}\right)=3, f\left(\dfrac{1}{e^{2}}\right)=\ln 6 . }$ Tính giá trị biểu thức $\Large\mathrm{f\left(\dfrac{1}{e}\right)+f\left(e^{3}\right) }$

• A. $\Large\mathrm{3(\ln 2+1) }$
• B. $\Large\mathrm{2 \ln 2}$
• C. $\Large\mathrm{3 \ln 2+1}$
• D. $\Large\mathrm{\ln 2 + 3}$

Từ $\Large\mathrm{f^{\prime}(x)=\dfrac{1}{x(\ln x-1)} \Rightarrow f(x)=\int \dfrac{1}{x(\ln x-1)} d x=\int \dfrac{d(\ln x-1)}{(\ln x-1)} }$ $\Large\mathrm{=\ln |\ln x-1|+C=\left\{\begin{array}{l}\ln (1-\ln x)+C_{1} \text { khi } x \in(0 ; e) \\ \ln (\ln x-1)+C_{2} \text { khi } x \in(e ;+\infty)\end{array}\right. }$

Ta có: $\Large\mathrm{\left\{\begin{array}{l}f\left(\dfrac{1}{e^{2}}\right)=\ln 6 \\ f\left(e^{2}\right)=3\end{array} \Leftrightarrow\left\{\begin{array}{l}\ln \left(1-\ln \dfrac{1}{e^{2}}\right)+C_{1}=\ln 6 \\ \ln \left(\ln e^{2}-1\right)+C_{2}=3\end{array} \Leftrightarrow\left\{\begin{array}{l}C_{1}=\ln 2 \\ C_{2}=3\end{array}\right.\right.\right. }$

Suy ra: $\Large\mathrm{f(x)=\left\{\begin{array}{l}\ln (1-\ln x)+\ln 2 \text { khi } x \in(0 ; e) \\ \ln (\ln x-1)+3 \quad \text { khi } x \in(e ;+\infty)\end{array}\right.}$$\Large \Rightarrow \mathrm{f\left(\dfrac{1}{e}\right)+f\left(e^{3}\right)=2 \ln 2+\ln 2+3=3(\ln 2+1)}$ $\Large\Rightarrow$ đáp án A