From: Vladimir Bondarenko on
In[1] := $Version

Out[1] = 7.0 for Microsoft Windows (32-bit) (February 18, 2009)

In[2] := Integrate[Cos[Pi z] Cos[z]^Pi, {z, 0, Pi/2}]

Out[2] = 0

In[3] := NIntegrate[Cos[Pi z] Cos[z]^Pi, {z, 0, Pi/2}]

Out[3] = 0.177994