Answer
$$\sin 2x - 2x\cos 2x + C$$
Work Step by Step
$$\eqalign{
& \int {4x} \sin 2xdx \cr
& {\text{write the integrand as}} \cr
& = \int {2x} \sin 2x\left( 2 \right)dx \cr
& {\text{Make an appropiate }}u{\text{ - substitution }} \cr
& u = 2x,\,\,\,\,\,\,\,du = 2dx,\,\,\,\,\, \cr
& {\text{write in terms of }}u \cr
& \int {2x} \sin 2x\left( 2 \right)dx = \int u \sin udu \cr
& {\text{Use the Endpaper Integral Table to evaluate the integral}} \cr
& {\text{By formula 44}} \cr
& \left( {44} \right):\,\,\,\,\,\,\int {u\sin udu = } \sin u - u\cos u + C \cr
& \cr
& {\text{write in terms of }}x{\text{; replace }}2x{\text{ for }}u \cr
& = \sin 2x - 2x\cos 2x + C \cr} $$
