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