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  restart; interface(version);

    Classic Worksheet Interface, Maple 12.00, Windows, Apr 10 2008 Build ID 347164

  Int(1/(u^2+1/4)*exp(I*u*k),u=-infinity...infinity); value(%);

                                  0

but

  Int(1/(u^2+1/4)*exp(I*u*k),u=-infinity...infinity); 
  convert(%,trig); value(%); simplify(%) assuming k::real;

                         2 Pi exp(-1/2 | k |)


For u = 0 this is a classical result I think:

  Int(1/(u^2+1/4)*exp(I*u*k),u=-infinity...infinity); 
  subs(k=0,%); value(%);

                                 2 Pi
convert infinity interface

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