`int(1)/(4+5 sin x)dx` का मान ज्ञात कीजिए ।

Explanation

`int(1)/(4+5sinx)dx`
`=int(1)/(4(sin^(2).(x)/(2)+cos^(2).(x)/(2))+5.2sin.(x)/(2)cos.(x)/(2))dx`
`=(1)/(2)int(sec^(2).(x)/(2))/(2 tan^(2).(x)/(2)+2+5tan.(x)/(2))dx`
(अंश और हर को `cos^(2).(x)/(2)` से भाग देने पर )
`int(dt)/(2t^(2)+5t+2)" माना "tan.(x)/(2)=t`
`=int(1)/((t+2)(2t+1))dt" "rArr (1)/(2)sec^(2).(x)/(2)dx=dt` ltBrgt `=(2)/(3) int(1)/(2t+1)dt -(1)/(3)int(1)/(t+2)dt`
(आंशिक भिन्न में बदलने पर )
`=(1)/(3)log|2t+1|-(1)/(3)log|t+2|+c`
`=(1)/(3)log|(2t+1)/(t+2)|+c`
`=(1)/(3)log|(2tan.(x)/(2)+1)/(tan.(x)/(2)+2)|+c`