Question

If $b>a$ and $I={\int }_a^b\frac{dx}{\sqrt{(x-a)(b-x)}}$ then $I$ equals

• A. $\pi /2$
• B. $\pi$
• C. $3\pi /2$
• D. $2\pi$

Answer 1

Answer: (B)

$\pi$

Substitute $t=\frac{1}{2}(x-a+x-b)=x-\frac{1}{2}(a+b),$ so that
$(x-a)(b-x)=(t+c)(c-t)=c^2-t^2$
where $c=\frac{1}{2}(b-a).$
Thus,
$I={\int }_{-c}^c\frac{dx}{\sqrt{(c^2-t^2)}}$
$=2{\int }_0^c\frac{dx}{\sqrt{(c^2-t^2)}}=2{\sin }^{-1}{\left(\frac{t}{c}\right)]}_0^c$
$=2\left[{\sin }^{-1}\right(1)-0]=\pi$