What is `int_(0)^(pi) (dx)/(1+2 sin^(2) x)` equal to ?

Options

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

Answer

C

Explanation

`I = int_(0)^(pi) (dx)/(1+2sin^(2)x) = 2int_(0)^(pi//2) (dx)/(1+2 sin^(2)x)`
`= 2int_(0)^(pi//2) (sec^(2)xdx)/(sec^(2) x+ 2 tan^(2)x) = 2 int_(0)^(pi//2) (sec^(2) xdx)/(1+3 tan^(2)x)`
Put `tan x = t rArr sec^(2) xdx = dt`
When x = 0 ,t = 0
When `x = (pi)/2 , t = oo`
`:. I = 2 int_(0)^(oo) (dt)/(1+3t^(2)) = 2/3int_(0)^(oo) (dt)/(t^(2) + (1/(sqrt(3)))^(2))`
`= 2/3 xx sqrt(3) [tan^(-1) (t)/(1//sqrt(3))]_(0)^(oo)`
`= 2/(sqrt(3))[tan^(-1) sqrt(3)t]_(0)^(oo)`
`= 2/(sqrt(3)) [tan^(-1) oo - tan^(-1) 0]`
`= 2/(sqrt(3)) xx (pi)/(2) = (pi)/(sqrt(3))`