x-xcosxsinx-x cosxdx

We can simplify the expression by factoring out x in the numerator and denominator:

(x(1-cosx))/(sinx(1-cosx))dx

Now we can cancel out the (1-cosx) terms:

x/sinx dx

Using substitution, we can let u = sinx and du = cosx dx:

∫x/u du

Integrating, we get:

x ln|u| + C

Substituting back in for u, we get the final answer:

x ln|sinx| + C