Linear First Order ODE/(1 + x^2) dy + 2 x y dx = cotangent x dx

Theorem
The linear first order ODE:
 * $(1): \quad \paren {1 + x^2} \rd y + 2 x y \rd x = \cot x \rd x$

has the general solution:
 * $y = \dfrac {\map \ln {\sin x} } {1 + x^2} + \dfrac C {1 + x^2}$

Proof
$(1)$ can be written as:
 * $(2): \quad \paren {1 + x^2} \dfrac {\rd y} {\rd x} + 2 x y = \cot x$

We notice straight away that:
 * $\dfrac {\rd} {\rd x} \paren {1 + x^2} = 2 x$

and so:
 * $\dfrac {\rd} {\rd x} \paren {1 + x^2} y = \cot x$

Thus the general solution becomes:

or:
 * $y = \dfrac {\map \ln {\sin x} } {1 + x^2} + \dfrac C {1 + x^2}$