Question

`int 1/(sin x - sqrt3 cos x) dx`

Answer 1

Given I = `int 1/(sin x - sqrt3 cos x) dx`

Let 1 = r cos θ and √3 = r sin θ

r = `sqrt(3 + 1) = 2`

And tan θ = √3 → θ = `pi/3`

=> `int 1/(sin x - sqrt3 cos x) dx = int 1/(rcos theta sin x - r sin theta cos x) dx`

= `1/r int 1/(sin (x - theta))dx`

= `1/r int cosec(x - theta)dx`

We know that `int cosec x  dx = log|tan (x/2 - pi/6)| + c`

`1/2 log |tan(x/2 - pi/6)| + c`

∴ I = `int 1/(sinx - sqrt3 cos x) dx`

`1/2 log |tan (x/2 - pi/6)| + c`