`int_(0)^(pi//2)(dx)/(a^(2)cos^(2)x+b^(2)sin^(2)x)` का मान ज्ञात कीजिए ।

Options

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

Answer

A

Explanation

`int_(0)^(pi//2)(1)/(a^(2)cos^(2)x+b^(2)sin^(2)x)dx`
`=int_(0)^(pi//2)(sec^(2)x)/(a^(2)+b^(2)tan^(2)x)dx`
(अंश और हर को `cos^(2)x` से भाग देने पर )
`=int_(0)^(oo)(dt)/(b(a^(2)+t^(2)))" "{:("माना "b tan x=t),(rArr" "b sec^(2)xdx=dt),(rArr" "sec^(2)xdx=(dt)/(b)):}`
`=(1)/(b).(1)/(a)[tan^(-1).(t)/(a)]_(0)^(oo)" "{:(x=0 ," पर ",t=0),(x=(pi)/(2)," पर ",t=oo):}`
`=(1)/(ab)[tan^(-1)oo-tan^(-1)0]`
`=(1)/(ab).(pi)/(2)=(pi)/(2ab)`