Unexpected Integrate::idiv : (Warning message)

• To: mathgroup at smc.vnet.net
• Subject: [mg73779] Unexpected Integrate::idiv : (Warning message)
• From: "dimitris" <dimmechan at yahoo.com>
• Date: Wed, 28 Feb 2007 04:33:26 -0500 (EST)

```In[2]:=
\$Version

Out[2]=
"5.2 for Microsoft Windows (June 20, 2005)"

In[3]:=
Clear["Global`*"]

In[4]:=
Integrate[Cos[a*x]*CosIntegral[b*x], {x, 0, Infinity}]
Integrate::idiv: Integral of Cos[a\x]\CosIntegral[b\x] does not
converge on {0, =E2=88=9E}.
Out[4]=
Integrate[Cos[a*x]*CosIntegral[b*x], {x, 0, Infinity}]

However

In[9]:=
(Integrate[Cos[#1[[1]]*x]*CosIntegral[#1[[2]]*x], {
x, 0, Infinity}] & ) /@ {{3, 2}, {2, 2}, {2, 3}}
N@%

Out[9]=
{-(Pi/6), -(Pi/8), 0}
Out[10]=
{-0.523599,-0.392699,0.}

results which agree with Gradshteyn and Ryzhik formula.

(Integrate[ CosIntegral[a x] Cos[b x], {x,0,Infinity} ]  is equal to
-(Pi/(2 a)) if a^2>b^2; zero if a^2<b^2; -(Pi/(4 p)) if a^2=b^2)

Interestingly the following setting does not work

In[10]:=
Integrate[Cos[a*x]*CosIntegral[b*x], {x, 0, Infinity},
GenerateConditions -> False]
Out[10]=
Integrate[Cos[a*x]*CosIntegral[b*x], {x, 0, Infinity},
GenerateConditions -> False]

More interestingly the following setting work in desirable way

In[98]:=
Integrate[Cos[a*x]*CosIntegral[b*x], x]
FullSimplify[%]
FullSimplify[Limit[%, x -> Infinity, Assumptions -> a > 0 && b > 0] -
Limit[%, x -> 0, Assumptions -> a > 0 && b > 0]]
{(FullSimplify[#1, a > b] & )[%], (FullSimplify[#1, a < b] & )[%]}
Integrate[Cos[a*x]*CosIntegral[a*x], x]
Limit[%, x -> Infinity, Assumptions -> a > 0] - Limit[%, x -> 0,
Assumptions -> a > 0]

Out[98]=
(I*(-ExpIntegralEi[(-I)*a*x - I*b*x] + ExpIntegralEi[I*a*x - I*b*x] -
ExpIntegralEi[(-I)*a*x + I*b*x] +
ExpIntegralEi[I*a*x + I*b*x]))/(4*a) +
(CosIntegral[b*x]*Sin[a*x])/a
Out[99]=
-((1/(4*a))*(I*(ExpIntegralEi[(-I)*(a - b)*x] - ExpIntegralEi[I*(a -
b)*x] + ExpIntegralEi[(-I)*(a + b)*x] -
ExpIntegralEi[I*(a + b)*x] + 4*I*CosIntegral[b*x]*Sin[a*x])))
Out[100]=
-((Pi*(1 + Sign[a - b]))/(4*a*Sign[a - b]))
Out[101]=
{-(Pi/(2*a)), 0}
Out[102]=
(CosIntegral[a*x]*Sin[a*x])/a - SinIntegral[2*a*x]/(2*a)
Out[103]=
-(Pi/(4*a))

Dimitris

```

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