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Câu hỏi: Họ tất cả các ngyên hàm của hàm số $f\left( x \right)=\dfrac{2x-1}{{{\left( x+1 \right)}^{2}}}$ trên khoảng $\left( -1;+\infty \right)$ là
A. $2\ln \left( x+1 \right)+\dfrac{2}{x+1}+C.$
B. $2\ln \left( x+1 \right)+\dfrac{3}{x+1}+C.$
C. $2\ln \left( x+1 \right)-\dfrac{2}{x+1}+C.$
D. $2\ln \left( x+1 \right)-\dfrac{3}{x+1}+C.$
A. $2\ln \left( x+1 \right)+\dfrac{2}{x+1}+C.$
B. $2\ln \left( x+1 \right)+\dfrac{3}{x+1}+C.$
C. $2\ln \left( x+1 \right)-\dfrac{2}{x+1}+C.$
D. $2\ln \left( x+1 \right)-\dfrac{3}{x+1}+C.$
Ta có $\int{f\left( x \right)dx=\int{\dfrac{2x-1}{{{\left( x+1 \right)}^{2}}}}}dx=\int{\dfrac{2\left( x+1 \right)-3}{{{\left( x+1 \right)}^{2}}}dx}=\int{\left[ \dfrac{2}{x+1}-\dfrac{3}{{{\left( x+1 \right)}^{2}}} \right]dx}=2\ln \left( x+1 \right)+\dfrac{3}{x+1}+C.$
Chú ý: $\int{\dfrac{1}{{{\left( ax+b \right)}^{2}}}dx}=-\dfrac{1}{a}.\dfrac{1}{ax+b}+C.$
Chú ý: $\int{\dfrac{1}{{{\left( ax+b \right)}^{2}}}dx}=-\dfrac{1}{a}.\dfrac{1}{ax+b}+C.$
Đáp án B.