If `bgta,` then `int_(a)^(b)(dx)/(sqrt((x-a)(b-x)))` is equal to

Options

A. `(pi)/(2)`
B. `pi`
C. `(pi)/(2)(b-a)`
D. `(pi)/(4)(b-a)`

Answer

B

Explanation

`int_(a)^(b)(dx)/(sqrt((x-a)(b-x)))`
`=int_(a)^(b)(1)/(sqrt(x^(2)+(a+b)x-ab))dx`
`=int_(a)^(b)(1)/(sqrt(((b-a)/(2))^(2)-(x-(a+b)/(2))^(2)))dx`
`=["sin"^(-1)((x-(a+b)/(2))/((b-a)/(2)))]_(a)^(b)`
`="sin"^(-1)(1)-sin^(-1)(-1)`
`=(pi)/(2)+(pi)/(2)=pi`