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Câu hỏi: Họ tất cả các nguyên hàm của hàm số $f\left( x \right)=\dfrac{3x-2}{{{\left( x-2 \right)}^{2}}}$ trên khoảng $\left( 2;+\infty \right)$ là
A. $3\ln \left( x-2 \right)+\dfrac{4}{x-2}+C.$
B. $3\ln \left( x-2 \right)+\dfrac{2}{x-2}+C.$
C. $3\ln \left( x-2 \right)-\dfrac{2}{x-2}+C.$
D. $3\ln \left( x-2 \right)-\dfrac{4}{x-2}+C.$
A. $3\ln \left( x-2 \right)+\dfrac{4}{x-2}+C.$
B. $3\ln \left( x-2 \right)+\dfrac{2}{x-2}+C.$
C. $3\ln \left( x-2 \right)-\dfrac{2}{x-2}+C.$
D. $3\ln \left( x-2 \right)-\dfrac{4}{x-2}+C.$
Ta có $f\left( x \right)=\dfrac{3x-2}{{{\left( x-2 \right)}^{2}}}=\dfrac{3\left( x-2 \right)+4}{{{\left( x-2 \right)}^{2}}}=\dfrac{3}{x-2}+\dfrac{4}{{{\left( x-2 \right)}^{2}}}$
Do đó $\int{\dfrac{3x-2}{{{\left( x-2 \right)}^{2}}}dx}=\int{\left( \dfrac{3}{x-2}+\dfrac{4}{{{\left( x-2 \right)}^{2}}} \right)dx}=3\ln \left( x-2 \right)-\dfrac{4}{x-2}+C$
Do đó $\int{\dfrac{3x-2}{{{\left( x-2 \right)}^{2}}}dx}=\int{\left( \dfrac{3}{x-2}+\dfrac{4}{{{\left( x-2 \right)}^{2}}} \right)dx}=3\ln \left( x-2 \right)-\dfrac{4}{x-2}+C$
Đáp án D.