Question

Gỉa sử \(\int\limits^5_3\dfrac{dx}{x^2-x}=aIn5+bIn3+cIn2\times\left(a,b,c\varepsilon Z\right)\) Tính giá trị biểu thức S\(-2a+b+3c^2\)

A. S= 3

B. S= 6

C. S= 0

D. S= -2

Answer 1

\(\int\limits^5_3\dfrac{dx}{x^2-x}=\int\limits^5_3\dfrac{dx}{x\left(x-1\right)}=\int\limits^5_3\left(\dfrac{1}{x-1}-\dfrac{1}{x}\right)dx\)

\(=\left(ln\left(x-1\right)-lnx\right)|^5_3=ln4-ln5-\left(ln2-ln3\right)\)

\(=2ln2-ln5-ln2+ln3\)

\(=-ln5+ln3+ln2\)

\(\Rightarrow a=-1;b=1;c=1\)