Question

$\int_{2}^{3}{\frac{dx}{x^{2} - x} =}$

• A. $\log(2/3)$
• B. $\log(1/4)$
• C. $\log(4/3)$
• D. $\log(8/3)$

Answer 1

Answer: (C)

$\log(4/3)$

Answer 2

$I = \int_{2}^{3}\frac{dx}{x^{2} - x}\mathbf{=}\int_{\mathbf{2}}^{\mathbf{3}}\frac{\mathbf{dx}}{\mathbf{x(x - 1)}}\mathbf{=}\int_{\mathbf{2}}^{\mathbf{3}}\left\lbrack \frac{\mathbf{1}}{\mathbf{x}\mathbf{-}\mathbf{1}}\mathbf{-}\frac{\mathbf{1}}{\mathbf{x}} \right\rbrack\mathbf{dx}$

$\mathbf{=}\int_{\mathbf{2}}^{\mathbf{3}}\frac{\mathbf{1}}{\mathbf{(x - 1)}}\mathbf{dx -}\int_{\mathbf{2}}^{\mathbf{3}}{\frac{\mathbf{1}}{\mathbf{x}}\mathbf{dx}}\mathbf{= \lbrack}\mathbf{\log}\mathbf{(}\mathbf{x}\mathbf{-}\mathbf{1)}\mathbf{\rbrack}_{\mathbf{2}}^{\mathbf{3}}\mathbf{-}\mathbf{\lbrack}\mathbf{\log}\mathbf{x}\mathbf{\rbrack}_{\mathbf{2}}^{\mathbf{3}}$

$\mathbf{= \lbrack}\mathbf{\log}\mathbf{2}\mathbf{-}\mathbf{\log}\mathbf{1}\mathbf{\rbrack - \lbrack}\mathbf{\log}\mathbf{3}\mathbf{-}\mathbf{\log}\mathbf{2}\mathbf{\rbrack}\mathbf{= 2}\mathbf{\log}\mathbf{2}\mathbf{-}\mathbf{\log}\mathbf{3}\mathbf{=}\mathbf{\log}\frac{\mathbf{4}}{\mathbf{3}}$.