If `int_(0)^(a) sqrt((a-x)/(x)) dx = (k)/(2)` ,then k = …..

Options

A. `(pia)/(2)`
B. `(5 pia)/(2)`
C. `(3 pi a)/(2)`
D. `pi a`

Answer

D

Explanation

We have `int_(0)^(a) sqrt((a - x)/(x)) dx = (k)/(2)`
Let `I = int_(0)^(a) sqrt((a - x)/(x)) dx`
Put `x = a "sin"^(2) theta`
`implies dx = a (2 "sin" theta cos theta) d theta`
When, `x = 0, theta = 0` and `x = a, theta = (pi)/(2)`
`:. i = int_(0)^(2) sqrt((a - a "sin"^(2) theta)/(a "sin"^(2) theta)) (2 a "sin" theta cos theta) d theta`
`= 2a int_(0)^(pi//2) (cot theta) "sin" theta cos theta d theta`
`= 2 a int_(0)^(pi//2) cos^(2) theta d theta = 2a int_(0)^(pi//2) (1 + cos 2 theta)/(2) d theta`
`= a int_(0)^(pi//2) (1 + cos 2 theta) d theta`
`= a[(theta + ("sin"2 theta)/(2))]_(0)^(pi//2) = a [(pi)/(2) + ("sin" pi)/(2)] = (pi a)/(2)`
`:. k = pi a`