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