Integration Bug in Mathematica

• To: mathgroup at christensen.cybernetics.net
• Subject: [mg589] Integration Bug in Mathematica
• From: Rod Frehlich <rgf at terra.colorado.edu>
• Date: Wed, 22 Mar 1995 07:28:19 -0700

```Integration bug in Mathematica for
Integrate[x Exp[-a x^2 ] Sin[b x + c]^2,{x,-Infinity,Infinity}]

Mathematica 2.2 for IBM RISC System/6000
-- Terminal graphics initialized --

In[3]:=
In[3]:= y = x Exp[-a x^2 ] Sin[b x + c]^2

2
x Sin[c + b x]
Out[3]= ---------------
2
a x
E

In[4]:=
In[4]:= y1 = Integrate[y,{x,-Infinity,Infinity}]

Out[4]= 0

In[5]:=
In[5]:= y = y /. {Sin[x_]^2 -> 1/2 - 1/2 Cos[2 x]}

1   Cos[2 (c + b x)]
x (- - ----------------)
2          2
Out[5]= ------------------------
2
a x
E

In[6]:=
In[6]:= y2 = Integrate[y,{x,-Infinity,Infinity}]

2
Sqrt[b ] Sqrt[Pi] Sin[2 c]
Out[6]= --------------------------
2
3/2  b /a
2 a    E

In[7]:=
In[7]:= (* y1 and y2 should be equal! y1 is obviously incorrect since the
integrand x Exp[-a x^2 ] Sin[b x + c]^2 is not an odd function *)

In[8]:=