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Câu hỏi: . Biết $\int{\dfrac{x+1}{\left( x-1 \right)\left( x-2 \right)}dx=a\ln \left| x-1 \right|}+b\ln \left| x-2 \right|+C,\left( a,b\in \mathbb{R} \right).$ Tính giá trị của biểu thức $a+b$.
A. $a+b=1.$
B. $a+b=5.$
C. $a+b=-5.$
D. $a+b=-1.$
A. $a+b=1.$
B. $a+b=5.$
C. $a+b=-5.$
D. $a+b=-1.$
$\int{\dfrac{x+1}{\left( x-1 \right)\left( x-2 \right)}dx}=\int{\dfrac{-2\left( x-2 \right)+3\left( x-1 \right)}{\left( x-1 \right)\left( x-2 \right)}dx}$
$=\int{\left( \dfrac{-2}{x-1}+\dfrac{3}{x-2} \right)dx}$
$=-2\ln \left| x-1 \right|+3\ln \left| x-2 \right|+C$
$\Rightarrow a=-2,b=3\Rightarrow a+b=1$
$=\int{\left( \dfrac{-2}{x-1}+\dfrac{3}{x-2} \right)dx}$
$=-2\ln \left| x-1 \right|+3\ln \left| x-2 \right|+C$
$\Rightarrow a=-2,b=3\Rightarrow a+b=1$
Đáp án A.