Question

Evaluate : $\int \frac{1}{\sqrt{(x-1)(x-2)}}dx.$

• A. $=\log |(x-\frac{3}{2})+\sqrt{x^2-3x+2}|+C$
• B. $=\log ((x-\frac{3}{2})+\sqrt{x^2-3x+2})+C$
• C. $=\log |(x-\frac{3}{2})+\sqrt{x^2-3x+2}|+5$
• D. $=\log |(x-\frac{3}{2})+\sqrt{x^2+3x+2}|+C$

Answer 1

The correct option is A $=\log |(x-\frac{3}{2})+\sqrt{x^2-3x+2}|+C$

$I=\int \frac{1}{\sqrt{x^2-3x+2}}dx$

$=\int \frac{1}{\sqrt{x^2-3x+\frac{9}{4}-\frac{9}{4}+2}}dx$

$=\int \frac{1}{\sqrt{{(x-\frac{3}{2})}^2-{\left(\frac{1}{2}\right)}^2}}dx$

$=\log |(x-\frac{3}{2})+\sqrt{{(x-\frac{3}{2})}^2-{\left(\frac{1}{2}\right)}^2}|+C$

$=\log |(x-\frac{3}{2})+\sqrt{x^2-3x+2}|+C$