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D

- Hướng dẫn giải

Chọn D

Đặt $\left\{\begin{array}{l}u=\ln (x+1)\\ dv=xdx\end{array}\Rightarrow \left\{\begin{array}{l}du=\frac{1}{x+1}dx\\ v=\frac{x^2-1}{2}\end{array}\right.\right.$

$$\begin{gathered} I={\int }_0^{e-1}x\ln (x+1)dx \\ =\left[\ln (x+1)\frac{x^2-1}{2}\right]{|}_0^{e-1}-\frac{1}{2}{\int }_0^{e-1}(x-1)dx \\ =\frac{e^2-2e}{2}-\frac{1}{2}\left(\frac{x^2}{2}-x\right){|}_0^{e-1} \\ =\frac{e^2-2e}{2}-\frac{1}{2}\frac{e^2-4e+3}{2}=\frac{e^2-3}{4} \end{gathered}$$