Integrate the following w.r.t.x. `(1)/(3+2sin x+cos x)`'

Explanation

`"Let I"=int (1)/(3+2sin x+cos x)dx`
Put tan (x/2)=t `therefore x=2 tan^(-1)t`
`therefore dx=(2dt)/(1+t^(2))and sin x=(2t)/(1+t^(2)), cos x=(1-t^(2))/(1+t^(2))`
`therefore I=int (1)/(3+2((2t)/(1+t^(2)))+((1-t^(2))/(1+t^(2)))).(2dt)/(1+t^(2))`
`=int (1+t^(2))/(3(1+t^(2))+4t+(1-t^(2))).(2dt)/(1+t^(2))`
`=2int (1)/(2t^(2)+4t+4)dt`
`=int (1)/(t^(2)+2t+1)dt`
`=int (1)/((t^(2)+2t+1)+1)dt=int (1)/((t+1)^(2)+(1)^(2))dt`
`=(1)/(1)tan^(-1)((t+1)/(1))+c =tan^(-1)[tan((x)/(2))+1]+c`