f[x_, r_] := x*(r - x)*Exp[-Pi*I/(2 r)*x^2];
g[x_, r_] := Exp[-Pi*I/(2 r)*x^2]*(r - Pi*I*x^2 - 2*x + Pi*I/r*x^3);

Clear[r];
FullSimplify[
  Integrate[f[x, r], {x, 0, r}] + 
   Integrate[-Sum[Sin[2*Pi*n*x]/(n*Pi), {n, 1, Infinity}]*g[x, r], {x,
      0, r}], r > 0 && Element[r, Integers]]  // N

r = 12; 
FullSimplify[
  Integrate[f[x, r], {x, 0, r}] + 
   Integrate[-Sum[Sin[2*Pi*n*x]/(n*Pi), {n, 1, Infinity}]*g[x, r], {x,
      0, r}], r > 0 && Element[r, Integers]]  // N


Output:
0.
2.64102 - 39.4256 I