I cannot wrap my head around this u substitution business, please help.

For example...

$\displaystyle \ln(\sin{x})=\int\ \frac{1}{\sin{x}}du$

$\displaystyle (u = \sin{x})$

The integral is taken with respect to u, not x, which is totally worthless to me. I could care less about u. I'm all about x. So how do I get the integral to be dx ?

other examples causing me trouble:

$\displaystyle \ln(e^{ix})=\int\ \frac{1}{e^{ix}}du$ (I need to show how this results in ix)

or

$\displaystyle \ln(\sqrt{1-x^2})=\int\ \frac{1}{\sqrt{1-x^2}}du$

Thanks