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Câu hỏi: Cho hàm số $f\left( x \right)$ thỏa mãn $x.{f}'\left( x \right)\ln x+f\left( x \right)=2{{x}^{2}}$, $\forall x\in \left( 1; +\infty \right)$ và $f\left( \text{e} \right)={{\text{e}}^{2}}$. Giá trị của tích phân $I=\int\limits_{e}^{{{e}^{2}}}{\dfrac{x}{f\left( x \right)}\text{d}x}$ bằng
A. $I=\dfrac{5}{3}$.
B. $I=2$.
C. $I=\dfrac{1}{2}$.
D. $I=\dfrac{3}{2}$.
A. $I=\dfrac{5}{3}$.
B. $I=2$.
C. $I=\dfrac{1}{2}$.
D. $I=\dfrac{3}{2}$.
Với $\forall x\in \left( 1; +\infty \right)$, ta có
$x.{f}'\left( x \right)\ln x+f\left( x \right)=2{{x}^{2}}$ $\Leftrightarrow {f}'\left( x \right)\ln x+f\left( x \right).\dfrac{1}{x}=2x$ $\Leftrightarrow {{\left( f\left( x \right).\ln x \right)}^{\prime }}=2x$ $\Rightarrow f\left( x \right).\ln x={{x}^{2}}+C$.
$\Rightarrow f\left( \text{e} \right).\ln \text{e}={{\text{e}}^{2}}+C$ $\Leftrightarrow {{\text{e}}^{2}}={{\text{e}}^{2}}+C$ $\Leftrightarrow C=0$.
$\Rightarrow f\left( x \right).\ln x={{x}^{2}}$ $\Rightarrow \dfrac{x}{f\left( x \right)}=\dfrac{\ln x}{x}$
$I=\int\limits_{\text{e}}^{{{\text{e}}^{2}}}{\dfrac{x}{f\left( x \right)}\text{d}x}$ $=\int\limits_{\text{e}}^{{{\text{e}}^{2}}}{\dfrac{\ln x}{x}\text{d}x}$ $=\int\limits_{\text{e}}^{{{\text{e}}^{2}}}{\ln x\text{d}\left( \ln x \right)=\left. \dfrac{{{\left( \ln x \right)}^{2}}}{2} \right|_{\text{e}}^{{{\text{e}}^{2}}}}$ $=\dfrac{{{\left( \ln {{\text{e}}^{2}} \right)}^{2}}}{2}-\dfrac{{{\left( \ln \text{e} \right)}^{2}}}{2}$ $=\dfrac{{{2}^{2}}}{2}-\dfrac{1}{2}=\dfrac{3}{2}$.
$x.{f}'\left( x \right)\ln x+f\left( x \right)=2{{x}^{2}}$ $\Leftrightarrow {f}'\left( x \right)\ln x+f\left( x \right).\dfrac{1}{x}=2x$ $\Leftrightarrow {{\left( f\left( x \right).\ln x \right)}^{\prime }}=2x$ $\Rightarrow f\left( x \right).\ln x={{x}^{2}}+C$.
$\Rightarrow f\left( \text{e} \right).\ln \text{e}={{\text{e}}^{2}}+C$ $\Leftrightarrow {{\text{e}}^{2}}={{\text{e}}^{2}}+C$ $\Leftrightarrow C=0$.
$\Rightarrow f\left( x \right).\ln x={{x}^{2}}$ $\Rightarrow \dfrac{x}{f\left( x \right)}=\dfrac{\ln x}{x}$
$I=\int\limits_{\text{e}}^{{{\text{e}}^{2}}}{\dfrac{x}{f\left( x \right)}\text{d}x}$ $=\int\limits_{\text{e}}^{{{\text{e}}^{2}}}{\dfrac{\ln x}{x}\text{d}x}$ $=\int\limits_{\text{e}}^{{{\text{e}}^{2}}}{\ln x\text{d}\left( \ln x \right)=\left. \dfrac{{{\left( \ln x \right)}^{2}}}{2} \right|_{\text{e}}^{{{\text{e}}^{2}}}}$ $=\dfrac{{{\left( \ln {{\text{e}}^{2}} \right)}^{2}}}{2}-\dfrac{{{\left( \ln \text{e} \right)}^{2}}}{2}$ $=\dfrac{{{2}^{2}}}{2}-\dfrac{1}{2}=\dfrac{3}{2}$.
Đáp án D.