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Câu hỏi: Cho hai hàm số f(x) và g(x) có đạo hàm trên $\mathbb{R}$ và thỏa mãn $f\left( 2 \right)+g\left( 2 \right)=5;g\left( x \right)=-x.{f}'\left( x \right);f\left( x \right)=-x.{g}'\left( x \right).$ Tính $I=\int\limits_{1}^{9}{\left[ f\left( x \right)+g\left( x \right) \right]dx}.$
A. $20\ln 3.$
B. $10\ln 3.$
C. $20\ln \dfrac{9}{2}.$
D. $10\ln \dfrac{9}{2}.$
A. $20\ln 3.$
B. $10\ln 3.$
C. $20\ln \dfrac{9}{2}.$
D. $10\ln \dfrac{9}{2}.$
Ta có $f\left( x \right)+g\left( x \right)=-x\left[ {f}'\left( x \right)+{g}'\left( x \right) \right]$
$\Rightarrow \int{\left[ f\left( x \right)+g\left( x \right) \right]dx}=-\int{x\left[ {f}'\left( x \right)+{g}'\left( x \right) \right]dx}=-\int{xd\left[ f\left( x \right)+g\left( x \right) \right]}$
$\Rightarrow \int{\left[ f\left( x \right)+g\left( x \right) \right]dx}=-x\left[ f\left( x \right)+g\left( x \right) \right]+C+\int{\left[ f\left( x \right)+g\left( x \right) \right]dx}$
$\Rightarrow x\left[ f\left( x \right)+g\left( x \right) \right]=C\Rightarrow f\left( x \right)+g\left( x \right)=\dfrac{C}{x}.$
Bài ra $f\left( 2 \right)+g\left( 2 \right)=5\Rightarrow 5=\dfrac{C}{2}\Rightarrow C=10\Rightarrow f\left( x \right)+g\left( x \right)=\dfrac{10}{x}$
$\Rightarrow I=\int\limits_{1}^{9}{\left[ f\left( x \right)+g\left( x \right) \right]dx}=\left. \int\limits_{1}^{9}{\dfrac{10}{x}dx}=10\ln \left| x \right| \right|_{1}^{9}=10\ln 9=20\ln 3.$
$\Rightarrow \int{\left[ f\left( x \right)+g\left( x \right) \right]dx}=-\int{x\left[ {f}'\left( x \right)+{g}'\left( x \right) \right]dx}=-\int{xd\left[ f\left( x \right)+g\left( x \right) \right]}$
$\Rightarrow \int{\left[ f\left( x \right)+g\left( x \right) \right]dx}=-x\left[ f\left( x \right)+g\left( x \right) \right]+C+\int{\left[ f\left( x \right)+g\left( x \right) \right]dx}$
$\Rightarrow x\left[ f\left( x \right)+g\left( x \right) \right]=C\Rightarrow f\left( x \right)+g\left( x \right)=\dfrac{C}{x}.$
Bài ra $f\left( 2 \right)+g\left( 2 \right)=5\Rightarrow 5=\dfrac{C}{2}\Rightarrow C=10\Rightarrow f\left( x \right)+g\left( x \right)=\dfrac{10}{x}$
$\Rightarrow I=\int\limits_{1}^{9}{\left[ f\left( x \right)+g\left( x \right) \right]dx}=\left. \int\limits_{1}^{9}{\dfrac{10}{x}dx}=10\ln \left| x \right| \right|_{1}^{9}=10\ln 9=20\ln 3.$
Đáp án A.