Simplification of a symbolic manipulation involving functions of more than one variable

Question

I am facing the following problem.

f[x_,y_] = a[x] u[y] + b[x] v[y]

Now I can ask Mathematica to calculate

g[x_] = Integrate[ Expand[f[x,y]^2],y]

which it correctly shows. In the expansion we have an expression like

Integrate[ a[x]^2 u[y]^2 + 2 a[x] b[x] u[v] v[y] + b[x]^2 v[y]^2 , y]

Now in the output I want to replace

Integrate[u[y]^2,y] 

by I11 etc. Using /. did not do the job. Can anyone please enlighten me?

Answer 1

The main difficulty here is that Integrate does not normally thread over addition. In other words, it does not understand that

$$\int f(x) + g(x) \text{d}x = \int f(x) \text{d}x + \int g(x) \text{d}x.$$

In most cases, it is true, but as integration is a limiting process care must be taken in the order of operations. A case where that is not true does not come to mind at the moment, other than involving infinite series, but it is not out of the question that a simplification in the integrand could obscure the relationship between the LHS and RHS, above. Mathematica, however, does understand

$$\int a(x) f(y) \text{d}y = a(x) \int f(y) \text{d}y,$$

as shown by

Integrate[a[x] u[y], y]
(*
 a[x] Integrate[u[y], y]
*)

A method to achieve the form you are looking for is to Map Integrate across the sum, Expand[f[x,y]^2], as follows

ints = Integrate[#, y]& /@ Expand[f[x,y]^2]
(*
  a[x]^2*Integrate[u[y]^2, y] + 2*a[x]*b[x]*Integrate[u[y]*v[y], y] 
+ b[x]^2*Integrate[v[y]^2, y]
*)

Then, ReplaceAll (/.) works just fine,

ints /. Integrate[u[y]^2, y] -> I11
(*
  I11*a[x]^2 + 2*a[x]*b[x]*Integrate[u[y]*v[y], y] 
+ b[x]^2*Integrate[v[y]^2, y]
*)