The value of `(dI)/(da)` when `I=int_(0)^(pi//2) log((1+asinx)/(1-asinx)) (dx)/(sinx)` (where `|a|lt1`) is

Options

A. `(pi)/(sqrt(1-a^(2)))`
B. `-pisqrt(1-a^(2))`
C. `sqrt(1-a^(2))`
D. `(sqrt(1-a^(2)))/(pi)`

Answer

A

Explanation

Let `I(a)=int_(0)^(pi//2) log((1+a sin x)/(1-a sinx))(dx)/(sinx)`
`(dI)/(da)=int_(0)^(pi//2) (2sinx)/(1-a^(2)sin^(2)x)(dx)/(x sin x)`
`=int_(0)^(pi//2)(2sec^(2)xdx)/(1+tan^(2)x-a^(2)tan^(2)x)`
`=int_(0)^(pi//2) (2sec^(2)xdx)/(1+tan^(2)x-a^(2)tan^(2)x)`
`=int_(0)^(oo) (2dt)/(1+(1-a^(2))t^(2))` (put `tan x=t`)
`=2/(sqrt(1-a^(2)))[tan^(-1)(tsqrt(1-a^(2)))]_(0)^(oo) =(pi)/(sqrt(1-a^(2))`
`:. I=pisin^(-1)a` [as `I(0)=0`]