Question

${\int }_a^b\sqrt{\left[\frac{x-a}{b-x}\right]}dx=$

• A. $\frac{2\pi (b-a)}{3}$
• B. $\frac{\pi (b+a)}{2}$
• C. $\frac{\pi (b-a)}{2}$
• D. $\frac{\pi (b-a)}{3}$

Answer 1

The correct option is C $\frac{\pi (b-a)}{2}$

${\int }_a^b\sqrt{\left[\frac{x-a}{b-x}\right]}dx={\int }_0^{\frac{\pi }{2}}\sqrt{\frac{(b-a)si{n}^2\theta }{(b-a)co{s}^2\theta }}(b-a)sin2\theta d\theta$
$=2.(b-a){\int }_0^{\frac{\pi }{2}}si{n}^2\theta d\theta =2(b-a).\frac{1}{2}.\frac{\pi }{2}=\frac{\pi (b-a)}{2}$