Is this a bug of mathematica 8?

Question

Clear["Global`*"]
Integrate[t f[x, y], {y, 0, 1}] - 
  t Integrate[f[x, y], {y, 0, 1}] // FullSimplify

Why doesn't M@ know the result is zero?

Answer 1

It is not a bug. Since your f[x,y] has no definition, Mathematica can't assume anything about the integrand t f[x, y]

You can make a rule to help Mathematica as mentioned below. But without a rule, Mathematica is doing the right thing here.

This has been discussed many places before. Here are some links

https://groups.google.com/forum/#!msg/comp.soft-sys.math.mathematica/jsiYo9tRj04/rQYCy-X3SXQJ

https://mathematica.stackexchange.com/questions/5610/how-to-simplify-symbolic-integration

For example, you can add this rule:

Clear["Global`*"]
Unprotect[Integrate];
Integrate[t_Symbol*f_,dom_]:=t*Integrate[f,dom];
Protect[Integrate];

Now it will give zero

Simplify@Integrate[t f[x,y],{y,0,1}]-t Integrate[f[x,y],{y,0,1}]
(*--->  0 *)