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Câu hỏi: Họ nguyên hàm của hàm số $f\left( x \right)=\left( 2x+5 \right)\ln x$ là:
A. $\left( {{x}^{2}}+5x \right)\ln x-\dfrac{{{x}^{2}}}{2}-5x+C$
B. $\left( {{x}^{2}}+5x \right)\ln x+\dfrac{{{x}^{2}}}{2}-5x+C$
C. $\left( {{x}^{2}}+5x \right)\ln x-\dfrac{{{x}^{2}}}{2}+5x+C$
D. $\left( {{x}^{2}}+5x \right)\ln x+\dfrac{{{x}^{2}}}{2}+5x+C$
A. $\left( {{x}^{2}}+5x \right)\ln x-\dfrac{{{x}^{2}}}{2}-5x+C$
B. $\left( {{x}^{2}}+5x \right)\ln x+\dfrac{{{x}^{2}}}{2}-5x+C$
C. $\left( {{x}^{2}}+5x \right)\ln x-\dfrac{{{x}^{2}}}{2}+5x+C$
D. $\left( {{x}^{2}}+5x \right)\ln x+\dfrac{{{x}^{2}}}{2}+5x+C$
Đặt $u=\ln x\to du=\dfrac{1}{2}dx$
$dv=\left( 2x+5 \right)dx\to v={{x}^{2}}+5x$
Suy ra $\int{\left( 2x+5 \right)\ln xdx}=\left( {{x}^{2}}+5x \right)\ln x-\int{\dfrac{{{x}^{2}}+5x}{x}dx}$
$=\left( {{x}^{2}}+5x \right)\ln x-\int{\left( x+5 \right)dx}=\left( {{x}^{2}}+5x \right)\ln x-\dfrac{{{x}^{2}}}{2}-5x+C$
$dv=\left( 2x+5 \right)dx\to v={{x}^{2}}+5x$
Suy ra $\int{\left( 2x+5 \right)\ln xdx}=\left( {{x}^{2}}+5x \right)\ln x-\int{\dfrac{{{x}^{2}}+5x}{x}dx}$
$=\left( {{x}^{2}}+5x \right)\ln x-\int{\left( x+5 \right)dx}=\left( {{x}^{2}}+5x \right)\ln x-\dfrac{{{x}^{2}}}{2}-5x+C$
Đáp án A.