Question

Integrate the following functions w.r.t. x : `int (1)/(4 - 5cosx).dx`

Answer 1

Let I = `int (1)/(4 - 5cosx).dx`

Put `tan(x/2)` = t
∴ x = 2 tan^–1 t

∴ dx = `(2dt)/(1 + t^2) and cosx = (1 - t^2)/(1 + t^2)`

∴ I = `int (1)/(4 - 5((1 - t^2)/(1 + t^2))).(2dt)/(1 + t^2)`

= `int (1 + t^2)/(4(1 + t^2) - 5(1 - t^2)).(2dt)/(1 + t^2)`

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

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

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

= `(2)/(9) xx (1)/(2 xx 1/3) log|(t - 1/3)/(t + 1/3)| + c`

= `(1)/(3) log |(3tan(x/2) - 1)/(3tan (x/2) + 1)| + c`.