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Câu hỏi: Họ nguyên hàm $\int\left(\dfrac{x^{2}+2 x+3}{x+1}\right) d x$ bằng
A. $\dfrac{{{x}^{2}}}{2}+x-2\ln |x+1|+C$.
B. $x^{2}+x+2 \ln |x+1|+C$.
C. $\dfrac{x^{2}}{2}+x-\dfrac{1}{(x+1)^{2}}+C$.
D. $\dfrac{x^{2}}{2}+x+2 \ln |x+1|+C$.
A. $\dfrac{{{x}^{2}}}{2}+x-2\ln |x+1|+C$.
B. $x^{2}+x+2 \ln |x+1|+C$.
C. $\dfrac{x^{2}}{2}+x-\dfrac{1}{(x+1)^{2}}+C$.
D. $\dfrac{x^{2}}{2}+x+2 \ln |x+1|+C$.
Ta có $\int\left(\dfrac{x^{2}+2 x+3}{x+1}\right) d x$ $=\int{\left( \dfrac{{{x}^{2}}+2x+1+2}{x+1} \right)}\text{d}x$
$=\int{\left( \dfrac{{{\left( x+1 \right)}^{2}}+2}{x+1} \right)}\text{d}x$ $=\int{\left( x+1+\dfrac{2}{x+1} \right)}\text{d}x$ $=\dfrac{{{x}^{2}}}{2}+x+2\ln \left| x+1 \right|+C$.
$=\int{\left( \dfrac{{{\left( x+1 \right)}^{2}}+2}{x+1} \right)}\text{d}x$ $=\int{\left( x+1+\dfrac{2}{x+1} \right)}\text{d}x$ $=\dfrac{{{x}^{2}}}{2}+x+2\ln \left| x+1 \right|+C$.
Đáp án D.