$\int\sin x\cos x\cot x\tan x\,dx$

Where the integrand is defined, \[ \cot x\tan x=1, \] so \[ \int \sin x\cos x\,dx =\boxed{\frac12\sin^2x+C.} \] This is an antiderivative on each open interval of the domain \(\{x:\sin x\cos x\neq0\}\) of the integrand.