start by putting the dy/dx by itself
[tex] \frac{dy}{dx}+xsin2y=x^3 cos^2y \\ \frac{dy}{dx}=x^3 cos^2y-xsin2y[/tex]
then multiply both sides by dx
[tex]dy=(x^3cos^2y-xsin2y)dx[/tex]
which you can now integrate both sides with respect to their variable, so the left by y and the right by x
[tex]y+ c_{1}= \frac{x^4}{4}cos^2y-\frac{x^2}{2}sin2y+c_{2} [/tex]
let c = c2-c1 and you find
[tex]y= \frac{x^4}{4}cos^2y-\frac{x^2}{2}sin2y+c [/tex]
