Google
Câu hỏi: Họ nguyên hàm của hàm số $f\left( x \right)=\dfrac{1}{x}\left( 2x-\ln x \right)$ là
A. $2x-\dfrac{{{\ln }^{2}}x}{2}+C$.
B. $2x-\dfrac{1}{{{x}^{2}}}+C$.
C. $\dfrac{2\ln \left| x \right|}{x}-\dfrac{1}{x}+C$.
D. $2x-\dfrac{\ln x}{x}+C$.
Ta có: $f\left( x \right)=\dfrac{1}{x}\left( 2x-\ln x \right)=2-\dfrac{1}{x}\ln x$
Suy ra $\int{f\left( x \right)\text{d}x=\int{2dx-\int{\dfrac{\ln x\text{d}x}{x}=2x-\int{\ln x\text{d}\left( \ln x \right)}}}}=2x-\dfrac{1}{2}{{\ln }^{2}}x+C$.
A. $2x-\dfrac{{{\ln }^{2}}x}{2}+C$.
B. $2x-\dfrac{1}{{{x}^{2}}}+C$.
C. $\dfrac{2\ln \left| x \right|}{x}-\dfrac{1}{x}+C$.
D. $2x-\dfrac{\ln x}{x}+C$.
Ta có: $f\left( x \right)=\dfrac{1}{x}\left( 2x-\ln x \right)=2-\dfrac{1}{x}\ln x$
Suy ra $\int{f\left( x \right)\text{d}x=\int{2dx-\int{\dfrac{\ln x\text{d}x}{x}=2x-\int{\ln x\text{d}\left( \ln x \right)}}}}=2x-\dfrac{1}{2}{{\ln }^{2}}x+C$.
Đáp án A.