`int(dx)/(2+cosx)` का मान ज्ञात कीजिए ।

Options

A. `(2)/(sqrt3)tan^(-1)((tan.(x)/(2))/(sqrt3))+c`
B. `(3)/(sqrt3)tan^(-1)((tan.(x)/(2))/(sqrt3))+c`
C. `(2)/(sqrt2)tan^(-1)((tan.(x)/(2))/(sqrt3))+c`
D. `(3)/(sqrt2)tan^(-1)((tan.(x)/(2))/(sqrt3))+c`

Answer

A

Explanation

`int(1)/(2+cosx)dx`
`" "int(1)/(2+2cos^(2).(x)/(2)-1)dx=int(1)/(1+2cos^(2).(x)/(2))dx`
`" "=int(1)/(sin^(2).(x)/(2)+cos^(2).(x)/(2)+2cos^(2).(x)/(2))dx`
`" "=int(1)/(sin^(2).(x)/(2)+3cos^(2).(x)/(2))dx`
`" "=int(sec^(2).(x)/(2))/(tan^(2).(x)/(2)+3)dx" माना "tan.(x)/(2)=t`
`=int(2dt)/(t^(2)+3)" "{:(rArr(1)/(2)sec^(2).(x)/(2)=(dt)/(dx)),(rArrsec^(2).(x)/(2)dx=2dt):}`
`" "=2int(1)/(t^(2)+(sqrt3)^(2))dt`
`" "=(2)/(sqrt3)tan^(-1)((t)/(sqrt3))+c`
`" "=(2)/(sqrt3)tan^(-1)((tan.(x)/(2))/(sqrt3))+c`