`int_(0)^(pi//2) ((nsqrt(sec x))/(nsqrt(sec x)+nsqrt(cosecx)))dx` is equal to

Options

A. `(pi)/(2)`
B. `(pi)/(3)`
C. `(pi)/(4)`
D. `(pi)/(6)`

Answer

C

Explanation

Let `=underset(0)overset((pi)/(2))int (nsqrt(sec x))/(nsqrt(sec x)+nsqrt(cosec x))dx.....(i)`
`=underset(0)overset((pi)/(2)) (nsqrt("cosec"x))/("cosec x")+nsqrt(secx)dx`......(ii)
On adding Eq. (i) and (ii), we get
`2l=underset(0)overset((pi)/(2))int ((nsqrt(sec x)+nsqrt("cosec"x)))/((nsqrt(sec x+nsqrt("cosec"x))))dx`
`Rightarrow 2l=underset(0)overset((pi)/(2))int dx=[x]_(0)^(pi//2)`
`Rightarrow 2l=(pi)/(2)`
`Rightarrow l=(pi)/(4)`