We have
[tex](-3xy^2+y)_y=--6xy+1[/tex]
and
[tex](-3x^2y+x)_x=-6xy+1[/tex]
so the equation is indeed exact. So we want to find a function [tex]F(x,y)=C[/tex] such that
[tex]F_x=-3xy^2+y[/tex]
[tex]F_y=-3x^2y+x[/tex]
Integrating both sides of the first equation wrt [tex]x[/tex] gives
[tex]F(x,y)=-\dfrac32x^2y^2+xy+f(y)[/tex]
Differentiating both sides wrt [tex]y[/tex] gives
[tex]F_y=-3x^2y+x=-3x^2y+x+f_y\implies f_y=0\implies f(y)=C[/tex]
So we have
[tex]F(x,y)=-\dfrac32x^2y^2+xy+C=C[/tex]
or
[tex]F(x,y)=\boxed{-\dfrac32x^2y^2+xy=C}[/tex]
