`int (dx)/(4 sin^(2)x+5cos^(2)x)` का मान है

Options

A. `(1)/(2)tan^(-1)(2tanx)+C`
B. `(1)/(2sqrt(5))"tan"^(-1)((2tanx)/(sqrt(5)))+C`
C. `(1)/(sqrt(5))"tan"^(-1)((tanx)/(sqrt(5)))+C`
D. इनमे से कोई नहीं

Explanation

`I= int(dx)/(4 sin^(2)x+5cos^(2)x)`
`= int (sec^(2)x dx)/(4 tan^(2)x+5)`
`= int (sec^(2)x dx)/((2 tan x)^(2)+(sqrt(5)^(2))dt`
माना ` tan x=t rArr sec(2)x dx =dt`
`:. I= (1)/(2)int (1)/(t^(2)+(sqrt(5)^(2)))dt`
`=(1)/(2sqrt(5))"tan^(-1)((t)/(sqrt(5)))+C`
`=(1)/(2sqrt(5))"tan^(-1)((2tanx)/(sqrt(5)))+C`