The following two codes give conflicting answers, when integrating sin(k*pi*x)*sin(2*n*pi*x) from 0 to 1, where both k and n are positive integers.

Code 1 assumes that k,n are independent integers:

    Integrate[Sin[k*Pi*x]*Sin[2*n*Pi*x],{x,0,1},
              Assumptions->{k\[Element]Integers,n\[Element]Integers}]

The result given by mathematica is 0.

Code 2 assumes that k=2*n and n is integer:

    Integrate[Sin[k*Pi*x]*Sin[2*n*Pi*x],{x,0,1},
              Assumptions->{n\[Element]Integers，k=2*n}]

The result is 1/2.

The result of Code 2 should be included in that of Code 1. It seems that Code 1 doesn't manage to give a general result.
Isn't Code 1 supposed to give a general result?