x-xcosxsinx-xcosxdx

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

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

Now we can cancel out the common factor of (1-cosx):

x/sinx dx

We can use u-substitution with u = sinx, du = cosx dx:

∫ x/sinx dx = ∫ du/u = ln|u| + C = ln|sinx| + C