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Câu hỏi: 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 $f\left( e \right)={{e}^{2}}$.
Tính tích phân $I=\int\limits_{e}^{{{e}^{2}}}{\dfrac{x}{f\left( x \right)}dx}$.
A. $\dfrac{3}{2}$.
B. $\dfrac{1}{2}$.
C. $\dfrac{5}{3}$.
D. 2.
Tính tích phân $I=\int\limits_{e}^{{{e}^{2}}}{\dfrac{x}{f\left( x \right)}dx}$.
A. $\dfrac{3}{2}$.
B. $\dfrac{1}{2}$.
C. $\dfrac{5}{3}$.
D. 2.
Ta có $x\ln x.{f}'\left( x \right)+f\left( x \right)=2{{x}^{2}}\Leftrightarrow \ln x.{f}'\left( x \right)+\dfrac{1}{x}.f\left( x \right)=2x$
$\Leftrightarrow {{\left( \ln x.f\left( x \right) \right)}^{\prime }}=2x\Leftrightarrow \ln x.f\left( x \right)=\int{2xdx}={{x}^{2}}+C$ mà $f\left( e \right)={{e}^{2}}\Rightarrow C=0$
Do đó $f\left( x \right)=\dfrac{{{x}^{2}}}{\ln x}\Rightarrow \dfrac{x}{f\left( x \right)}=\dfrac{\ln x}{x}\Rightarrow I=\left. \dfrac{{{\ln }^{2}}x}{2} \right|_{e}^{{{e}^{2}}}=\dfrac{3}{2}$.
$\Leftrightarrow {{\left( \ln x.f\left( x \right) \right)}^{\prime }}=2x\Leftrightarrow \ln x.f\left( x \right)=\int{2xdx}={{x}^{2}}+C$ mà $f\left( e \right)={{e}^{2}}\Rightarrow C=0$
Do đó $f\left( x \right)=\dfrac{{{x}^{2}}}{\ln x}\Rightarrow \dfrac{x}{f\left( x \right)}=\dfrac{\ln x}{x}\Rightarrow I=\left. \dfrac{{{\ln }^{2}}x}{2} \right|_{e}^{{{e}^{2}}}=\dfrac{3}{2}$.
Đáp án A.