How to operate with function body without evaluation

Question

I have several functions (let a[x] and b[x]), defined as expressions with integrals and differentiations. I want to perform some manipulations with these expressions - for example, ask Mathematica to simplify a[x] + b[x] without evaluations of integrals. Hold doesn't works (as I tried) as I used to expand functions a[x] and b[x] first. I think, that I could replace Integrate and D with some meaningless heads integrate and d, and then expand functions with another meaningless symbol x, but I can't achieve this, because Mathematica tries first to perform integrations and fails.

Example:

a[t_] := Integrate[t, {x, -∞, ∞}]
b[t_] := Integrate[t^2, {x, -∞, ∞}]
a[t]+b[t]  (* this doesn't works, as it "evaluates" integrals; 
              I want to see simple sum *)

ADDED: I want to see something like myIntegrate[t, {x, -∞, ∞}]+myIntegrate[t^2, {x, -∞, ∞}]. Really, a and b are more complicated, and they contain more than one integrals. I want Mathematica to perform expression simplification keeping integrals unevaluated. I could manually modify definition for a and b, but they are rather complicated, and don't want to keep two copies (one with Integrate, and one with some myIntegrate).

Answer 1

One way of doing this would be using Defer or HoldForm. For example, let us define the functions a and b as follows:

 a[t_] := Defer[Integrate[t, {x, -∞, ∞}]]
 b[t_] := Defer[Integrate[t^2, {x, -∞, ∞}]]

and

 a1[t_] := HoldForm[Integrate[Exp[-t], {x, 0, ∞}]]
 b1[t_] := HoldForm[Integrate[Exp[-t^2], {x, 0, ∞}]]

They both return unevaluated function. Then the function:

 mySum[expr1_, expr2_] := ReplacePart[expr1, {1, 1} -> expr1[[1, 1]] + expr2[[1, 1]]]

does the job.

mySum[a1[t], b1[t]]

The result looks as follows:

The same will take place with the functions a[t] and b[t]