A Frullani integral and some workouts

• To: mathgroup at smc.vnet.net
• Subject: [mg75289] A Frullani integral and some workouts
• From: dimitris <dimmechan at yahoo.com>
• Date: Tue, 24 Apr 2007 03:21:41 -0400 (EDT)

```In[75]:=
Integrate[(Exp[(-a)*y] - Exp[-2*a*y])/y, {y, 0, Infinity}, Assumptions
-> a > 0]
(*incorrect; buggy imaginary part*)

Out[75]=
I*Pi + Log[2]

Some workouts include

In[81]:=
Integrate[(Exp[(-a)*y] - Exp[-2*a*y])/y, y]
Assuming[a > 0, Limit[%, y -> Infinity] - Limit[%, y -> 0]]

Out[81]=
-ExpIntegralEi[-2*a*y] + ExpIntegralEi[(-a)*y]
Out[82]=
Log[2]

In[99]:=
Integrate[(Exp[(-a)*y] - Exp[-2*EulerGamma*y])/y, {y, 0, Infinity},
Assumptions -> a > 0]
% /. EulerGamma -> a

Out[99]=
Log[(2*EulerGamma)/a]
Out[100]=
Log[2]

In[108]:=
((Exp[(-a)*y] - Exp[-2*a*y])/y)*Dt[y] /. y -> o/a /. {Dt[a] -> 0,
Dt[o] -> 1}
Integrate[%, {o, 0, Infinity}]

Out[108]=
(-E^(-2*o) + E^(-o))/o
Out[109]=
Log[2]

Dimitris

```

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