For any function, f(x,y), of two variables, df= f_xdx+ f_ydy. In particular, if you are given that da= (3x^2y^2+ 8xy^3)dx+ (3x^3y+ 12x^2y^2+ 4y)dy then a_x= 3x^2y^2+ 8xy^3 and a_y= 2x^3y+ 12x^2y^2+ 4y.
I presume you know that, for any function, a(x,y), with continuous second derivatives, the two mixed second derivatives must be equal: a_{xy}= a_{yx}. We can use that to check if there is, in fact, a solution to this problem- in general, just putting to simple functions, say, f(x,y) and g(x,y) together as f(x,y)dx+ g(x,y) does NOT result in an "exact differential" because f_y\ne g_x. But here, a_{xy}= (3x^2y^2+ 8xy^3)_y= 6x^2y+ 24xy^2= (2x^3y+ 12x^2y^2+ 4y)_x.
So, knowing that a_x= 3x^2y^2+ 8xy^3 and that the partial derivative with respect to x treats y like a constant, we can find a by integrating with respect to x, treating y as a constant. HOWEVER, that means that the "constant" of integration might actually be a function of y- call that, say F(y). So now differentiate the "a" you just got with respect to y, leaving the derivative of F as F' and set that equal to 2x^3y+ 12x^2y^2+ 4y. Because of the check above, we know that the "x" terms will cancel leaving an expression, involving only y, for F'. Integrate that to find F.
(I am puzzled by your reference to "differential k forms". They usually arise in an advanced differential geometry course. But the problem you give is a standard "third semester Calculus" or "Calculus of functions of several variables" problem.)
