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Câu hỏi: Tích phân $\int\limits_{1}^{e}{\dfrac{\ln x}{x}dx}$ bằng
A. $\dfrac{1}{2}.$
B. $\dfrac{{{e}^{2}}-1}{2}.$
C. $\dfrac{{{e}^{2}}+1}{2}.$
D. $-\dfrac{1}{2}.$
A. $\dfrac{1}{2}.$
B. $\dfrac{{{e}^{2}}-1}{2}.$
C. $\dfrac{{{e}^{2}}+1}{2}.$
D. $-\dfrac{1}{2}.$
Ta có $\int\limits_{1}^{e}{\dfrac{\ln x}{x}dx}=\int\limits_{1}^{e}{\ln x.d\left( \ln x \right)=\dfrac{{{\ln }^{2}}x}{2}\left| _{\begin{smallmatrix}
\\
1
\end{smallmatrix}}^{\begin{smallmatrix}
e \\
\end{smallmatrix}} \right.=\dfrac{{{\ln }^{2}}e}{2}-\dfrac{{{\ln }^{2}}1}{2}=\dfrac{1}{2}.}$
\\
1
\end{smallmatrix}}^{\begin{smallmatrix}
e \\
\end{smallmatrix}} \right.=\dfrac{{{\ln }^{2}}e}{2}-\dfrac{{{\ln }^{2}}1}{2}=\dfrac{1}{2}.}$
Đáp án A.