Derive the integral of ∫$\csc(x)$dx using Differential Equation?

Question

I remember a while back someone showed me how to derive the $\int \csc xdx$ by differentiating another term. The derivative would contain a $\csc x$ which could be made the subject. Both sides could then be integrated to derive the integral. I have spent this morning trying to figure it out but can't for the life of me.

[EDIT]

I didn't clarify what I meant. I am aiming for something like this: e.g.

$\frac{\mathrm{d}\ (x csc x )}{\mathrm{d} x} = cosecx - xcosecxcotx$

$\frac{\mathrm{d}\ (x csc x )}{\mathrm{d} x} + xcosecxcotx = cosecx$

Integrating both sides

$x csc x + \int (xcosecxcotx) dx= \int (cosecx)dx $

Obviously this isn't the right term to begin with but I was hunting a method that is along these lines.

Answer 1

I would say:

$\int \csc x \ dx = \int \csc x \left( \frac {\csc x + \cot x}{\csc x + \cot x}\right) \ dx = \int \frac {\csc^2 x + \csc x\cot x}{\csc x + \cot x} \ dx\\ u = \csc x + \cot x\\ du = -\csc x\cot x - \csc^2 x \ dx\\ \int -\frac {1}{u} \ du\\ -\ln |\csc x + \cot x| + C$

And if you want to you could say:

$-\ln |\frac {1+\cos x}{\sin x}| + C\\ \ln |\frac {\sin x}{1+\cos x}| + C\\ \ln |\tan \frac 12 x| + C\\ $

Answer 2

$$ \frac {d}{dx} \ln (\csc x + \cot x) = \frac {-\csc x \cot x - \csc ^2 x}{\csc x + \cot x } = -\csc x $$ Thus, $$ \int \csc xdx= -ln | \csc x + \cot x | +C$$