`int_(0)^(pi//2)(dx)/(a^(2)cos^(2)x+b^(2)sin^(2)x)` किसके बराबर है ?

Options

A. 2ab
B. `2piab`
C. `pi//2ab`
D. `pi//ab`

Answer

C

Explanation

माना `I=int_(0)^(pi//2)(dx)/(a^(2)cos^(2)x+b^(2)sin^(2)x)`
`rArrI=int_(0)^(pi//2)(sec^(2)x)/(a^(2)+b^(2)tan^(2)x)dx`
[अंश व हर में`cos^(2)x` से भाग देने पर]
माना `tanx=trArrsec^(2)xdx=dt` जब x=0, तब t=0
जब `x=(pi)/(2)`, तब `t=infty`
`thereforeI=int_(0)^(infty)(dt)/(a^(2)+b^(2)t^(2))=(1)/(b^(2))int_(0)^(infty)(dt)/(((a)/(b))^(2)+t^(2))`
`=(1)/(b^(2))xx(1)/(((a)/(b)))[tan^(-1)((bt)/(a))]_(0)^(infty)`
`[becauseint(dx)/(a^(2)+x^(2))=(1)/(a)tan^(-1)((x)/(a))+C]`
`=(1)/(ab)[tan^(-1)(infty)-tan^(-1)(0)]`
`=(1)/(ab)((pi)/(2)-0)=(1)/(ab)xx(pi)/(2)=(pi)/(2ab)`