Evaluate : `int 1/(3+2 sin x + cos x) dx.`

Explanation

Let `I = int1/(3+2 sin x + cos x) dx`
Let `tan(x/2) = t`
`:. " " x = 2 tan^(-1) t`
` :. " " dx = (2 dt)/(1+t^(2)), sin x = (2t)/(1 + t^(2)),`
`cos x = (1-t^(2))/(1+t^(2))`
`:. " "I = int 1/(3 + 2 *(2t)/(1+t^(2))+(1-t^(2))/(1+t^(2)))*(2 dt)/(1+t^(2))`
` = int (2dt)/((3+3t^(2)+4t+1-t^(2))/((1+t^(2))))xx1/((1+t^(2)))`
` = int (2 dt)/(2t^(2) +_ 4t + 4)`
` = int (2 dt)/(2(t^(2) + 2t + 2))`
` = int(dt)/(t^(2)+2t+2)`
`= int (dt)/(t^(2) + 2t+1+2-1)`
` = int(dt)/((t+1)^(2)+1^(2))= tan^(-1) (t+1) + C.`
(C - constant of integration)
` = tan^(-1) [tan(x/2)+1] + C `