Question

Integrate the following functions w.r.t. x : `int (1)/(3 + 2sin x - cosx)dx`

Answer 1

Let I = `int (1)/(3 + 2sin x - cosx)dx`

Put `tan(x/2)` = t
∴ x = 2 tan^–1 t

∴ dx = `(2)/(1 + t^2)dt` and

sinx = `(2t)/(1 + t^2)' cosx = (1 - t^2)/(1 + t^2)`

∴ 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)`

= `2 int dt/(4t^2 + 4t + 2)`

= `2 int dt/(4t^2 + 4t + 1 + 1)`

= `2 int dt/((2t + 1)^2 + 1^2)`

= `(2)/(2)tan^-1((2t + 1)/1) + c`

= `tan^-1[2tan^-1(x/2) + 1] + c`.