`int_(0)^(pi//2)(dx)/((4sin^(2)x+5cos^(2)x))` का मान ज्ञात कीजिए ।

Explanation

`int_(0)^(pi//2)(dx)/((4sin^(2)x+5cos^(2)x))`
अंश व हर को `cos^(2)x` से भाग देने पर,
`int_(0)^(pi//2)(dx)/((4sin^(2)x+5cos^(2)x))=int_(0)^(pi//2)(sec^(2)x)/((4tan^(2)x+5))dx`
यदि `tanx=t rArr sec^(2) xdx = dt` व `x=0 rArr t = 0`
व `x=(pi)/(2)rArr t=oo,` तब
`=int_(0)^(oo)(dt)/((4t^(2)+5))`
`=(1)/(4)int_(0)^(oo)(dt)/(t^(2)+((sqrt5)/(2))^(2))=(1)/(4).(2)/(sqrt5)[tan^(-1).(2t)/(sqrt5)]_(0)^(oo)`
`=(1)/(2sqrt5[tan^(-1)(oo)-tan^(-1)(0)])`
`=(1)/(2sqrt5)((pi)/(2)-0)=(pi)/(4sqrt5)`