Question

Evaluate the following :
${\int }_2^3\frac{dx}{x^2-x}=$

• A. $\log \frac{2}{3}$
• B. $\log \frac{4}{3}$
• C. $\log \frac{8}{3}$
• D. $\log \frac{1}{2}$

Answer 1

The correct option is B $\log \frac{4}{3}$

${\int }_2^3\frac{dx}{x^2-x}$

$\int \frac{dx}{(x^2-x)}=\int \frac{dx}{x(x-1)}=\int \frac{\left(x\right)-(x-1)}{x(x-1)}dx$

$=\int (\frac{1}{x-1}-\frac{1}{x})dx=\int \frac{1}{x-1}dx-\int \frac{1}{x}dx$

$=\log (x-1)-\log \left(x\right)$

$=\log \left(\frac{x-1}{x}\right)+c$

${\int }_2^3\frac{dx}{x^2-x}={\left[\log \left(\frac{x-1}{x}\right)\right]}_2^3$

$=\left(\log \left(\frac{2}{3}\right)\right)-\left(\log \left(\frac{1}{2}\right)\right)$

$=\log \left(\frac{\frac{2}{3}}{\frac{1}{2}}\right)=\log \left(\frac{4}{3}\right)$