Question

Integrate the following functions w.r.t. x : `int (1)/(3 + 2 sin2x + 4cos 2x).dx`

Answer 1

Let I = `int (1)/(3 + 2 sin2x + 4cos 2x).dx`

Put tan x = t
∴ x = tan^–1 t

∴ dx = `dt/(1 + t^2) and sin 2x = (2t)/(1 + t^2),, cos2x = (1 - t^2)/(1 + t^2)`

∴ I = `int (1)/(3 + 2((2t)/(1 + t^2)) + 4((1 - t^2)/(1 + t^2))).dt/(1 + t^2)`

= `int (1 + t^2)/(3(1 + t^2) + 4t + 4(1 - t^2)).dt/(1 + t^2)`

= `int (1)/(7 + 4t - t^2)dt = int (1)/(7 - (t^2 - 4t + 4) + 4)dt`

= `int (1)/((sqrt(11))^2 - (t - 2)^2)dt`

= `(1)/(2sqrt(11))log|(sqrt(11) + t - 2)/(sqrt(11) - t + 2)| + c`

= `(1)/(2sqrt(11))log|(sqrt(11) + tan x - 2)/(sqrt(11) - tan x + 2)| + c`.