# GeneratedParameters in Integrate

I defined a function of two variables, let's say

u[x_, y_] := 4 x^4 + 6 x y - 24 x^2 y^2 + 4 y^4

I want to partial integrate (with respect to y) the partial derivative of u with respect to x obtaining a function v[x_,y_] in which is present an additive function of x, let's say c[x]. I tried

v[x_, y_] = Integrate[\!$$\*SubscriptBox[\(\[PartialD]$$, $$x$$]\ $$u[x, y]$$\), y,
GeneratedParameters -> C]


but so I obtain a constant function c_1. Otherwise I have to add to the integral c[x]. Is it possible to obtain an additive function of x with GeneratedParameters?

• Why not adding c[x] outside of integral? i.e. v[x_, y_] = Integrate[\!$$\*SubscriptBox[\(\[PartialD]$$, $$x$$]\ $$u[x, y]$$\), y] + c[x] Jul 24 at 19:01

c[x] is not possible because MMA may need several constants and these are written: c, c.. However, you may indicate that the constant depends on x by using GeneratedParameters -> C[x] what will result in: c[x] what is displayed as e.g. with your example:
u[x_, y_] := 4 x^4 + 6 x y - 24 x^2 y^2 + 4 y^4
Integrate[\!$$\*SubscriptBox[\(\[PartialD]$$, $$x$$]\ $$u[x, y]$$\), y, • E.g.: Integrate[\!$$\*SubscriptBox[\(\[PartialD]$$, $$x$$]\ $$u[x, y]$$\), y, GeneratedParameters -> C[x]]Integrate[\!$$\*SubscriptBox[\(\[PartialD]$$, $$x$$]\ $$u[x, y]$$\), y, GeneratedParameters -> C[x]] /. C[x] -> f[x] Jul 24 at 20:55