Can someone check my working here? thanks

form the corresponding analytic function f(z)= u+iv

$\displaystyle u=4xy^3 - 4x^3y+x$

$\displaystyle u_x=4y^3-12x^2y+1=v_y$

$\displaystyle v=\int4y^3-12x^2y+1\cdot dx $

$\displaystyle v=y^4-6y^2x^2+y+f(x)$

$\displaystyle v_x=-12xy^2 + f'(x) = -u_y= -12xy^2 + 4x^3$

$\displaystyle f'(x)= 4x^3$

$\displaystyle f(x)=x^4 + c$

$\displaystyle v=y^4 - 6y^2x^2 + y + x^4 +c$

$\displaystyle f(z) = 4xy^3 - 4x^3y+x + i(y^4 - 6y^2x^2 + y + x^4 +c)$