Question

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

Answer 1

Let I = `int (1)/(3 + 2sinx).dx`

Put `tan (x/2) = t`

∴ x = 2 tan^–1 t

∴ dx = `(2t)/(1 + t^2) and sinx = (2t)/(1 + t^2)`

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

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

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

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

= `(2)/(3) int (1)/((t^2 + 4/3t + 4/9) - (4)/(9) + 1)dt`

= `(2)/(3) int (1)/((t + 2/3)^2 + (sqrt(5)/3)^2)dt`

= `(2)/(3) xx (1)/((sqrt(5)/3))tan^-1 [(t + 2/3)/(sqrt(5)/(3))] + c`

= `(2)/sqrt(5)tan^-1 ((3t + 2)/sqrt(5)) + c`

= `(2)/sqrt(5)tan^-1 [(3tan(x/2) + 2)/sqrt(5)] + c`.