Proper way to simplify integral result in Mathematica given integer constraints

Question

Evaluating the following integral should be non-zero, and mathematica correctly gives a non-zero result

Integrate[ Cos[ (Pi * x)/2 ]^2 * Cos[ (3*Pi*x)/2 ]^2, {x,         


        

Answer 1

Not always zero ...

k = Integrate[
         Cos[(Pi x)/2]^2 Cos[((2 (n) + 1) Pi x)/2] Cos[((2 m + 1) Pi x)/ 2], 
         {x, -1, 1}, Assumptions -> Element[{m, n}, Integers]];

(*Let's find the zeroes of the denominator *)

d = Denominator[k];
s = Solve[d == 0, {m, n}]

(*The above integral is indeterminate at those zeroes, so let's compute 
  the integral again there (a Limit[] could also do the work) *)

denZ = Integrate[
          Cos[(Pi x)/2]^2 Cos[((2 (n) + 1) Pi x)/2] Cos[((2 m + 1) Pi x)/ 2] /.s, 
          {x, -1, 1}, Assumptions -> Element[{m, n}, Integers]];

(* All possible results are generated with m=1 *)

denZ /. m -> 1

(*
{1/4, 1/2, 1/4, 1/4, 1/2, 1/4}
*)

Visualizing those cases:

Plot[Cos[(Pi x)/2]^2 Cos[((2 (n) + 1) Pi x)/2] Cos[((2 m + 1) Pi x)/2] 
     /. s /. m -> 1, {x, -1, 1}]

Compare with a zero result integral one:

Plot[Cos[(Pi x)/2]^2 Cos[((2 (n) + 1) Pi x)/2] Cos[((2 m + 1) Pi x)/ 2] 
     /. {m -> 1, n -> 4}, {x, -1, 1}]