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

If you just drop the whole FullSimplify part, mathematica does the integration neatly for you.

Integrate[
 Cos[(Pi x)/2]^2 Cos[((2 n + 1) Pi x)/2] Cos[((2 m + 1) Pi x)/
    2], {x, -1, 1}]

To include the condition that m and n are integers, it's better to use the Assumptions option in Integrate.

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]]