Biết

B

- Hướng dẫn giải

Đáp án B

Ta có $\underset{1}{\overset{e}{\int }}\frac{\left(x+1\right)\ln x+2}{1+x\ln x}dx=\underset{1}{\overset{e}{\int }}\frac{1+x\ln x+1+\ln x}{1+x\ln x}dx=\underset{1}{\overset{e}{\int }}\left(1+\frac{1+\ln x}{1+x\ln x}\right)dx=\underset{1}{\overset{e}{\int }}dx+\underset{1}{\overset{e}{\int }}\frac{1+\ln x}{1+x\ln x}dx$

$=\left.x\right|{}_1{}^e+\underset{1}{\overset{e}{\int }}\frac{d\left(1+x\ln x\right)}{1+x\ln x}=e-1+\ln \left|1+x\ln x\right|\left|{}_1{}^e\right.=e-1+\ln \left(e+1\right)=e+\ln \left(\frac{e+1}{e}\right)\Rightarrow \left\{\begin{array}{l}a=1\\ b=1\end{array}\right..$