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Câu hỏi: Cho tích phân $I=\int\limits_{1}^{e}{{{x}^{2}}.{{\ln }^{2}}xdx}.$ Mệnh đề nào sau đây đúng?
A. $I=\left. {{x}^{3}}{{\ln }^{2}}x \right|_{1}^{e}-2\int\limits_{1}^{e}{{{x}^{2}}\ln xdx}.$
B. $I=\left. {{x}^{3}}{{\ln }^{2}}x \right|_{1}^{e}-\dfrac{2}{3}\int\limits_{1}^{e}{{{x}^{2}}\ln xdx}.$
C. $I=\left. \dfrac{1}{3}{{x}^{3}}{{\ln }^{2}}x \right|_{1}^{e}-\dfrac{2}{3}\int\limits_{1}^{e}{{{x}^{2}}\ln xdx}.$
D. $I=\left. \dfrac{1}{3}{{x}^{3}}{{\ln }^{2}}x \right|_{1}^{e}-4\int\limits_{1}^{e}{x\ln xdx}.$
A. $I=\left. {{x}^{3}}{{\ln }^{2}}x \right|_{1}^{e}-2\int\limits_{1}^{e}{{{x}^{2}}\ln xdx}.$
B. $I=\left. {{x}^{3}}{{\ln }^{2}}x \right|_{1}^{e}-\dfrac{2}{3}\int\limits_{1}^{e}{{{x}^{2}}\ln xdx}.$
C. $I=\left. \dfrac{1}{3}{{x}^{3}}{{\ln }^{2}}x \right|_{1}^{e}-\dfrac{2}{3}\int\limits_{1}^{e}{{{x}^{2}}\ln xdx}.$
D. $I=\left. \dfrac{1}{3}{{x}^{3}}{{\ln }^{2}}x \right|_{1}^{e}-4\int\limits_{1}^{e}{x\ln xdx}.$
Đặt $\left\{ \begin{aligned}
& u={{\ln }^{2}}x \\
& dv={{x}^{2}}dx \\
\end{aligned} \right.\Leftrightarrow \left\{ \begin{aligned}
& du=\dfrac{2\ln x}{x}dx \\
& v=\dfrac{{{x}^{3}}}{3} \\
\end{aligned} \right.. $ Khi đó $ I=\left. \dfrac{1}{3}{{x}^{3}}{{\ln }^{2}}x \right|_{1}^{e}-\dfrac{2}{3}\int\limits_{1}^{e}{{{x}^{2}}\ln xdx}.$
& u={{\ln }^{2}}x \\
& dv={{x}^{2}}dx \\
\end{aligned} \right.\Leftrightarrow \left\{ \begin{aligned}
& du=\dfrac{2\ln x}{x}dx \\
& v=\dfrac{{{x}^{3}}}{3} \\
\end{aligned} \right.. $ Khi đó $ I=\left. \dfrac{1}{3}{{x}^{3}}{{\ln }^{2}}x \right|_{1}^{e}-\dfrac{2}{3}\int\limits_{1}^{e}{{{x}^{2}}\ln xdx}.$
Đáp án C.