Mathematica 10.1.0 returns a strange result for the following double integral
f = Integrate[ 1/(1 + x^2 y^2), {x, 0, ∞}, {y, 0, ∞}]
(* Out[1407]= \!\(
\*SubsuperscriptBox[\(∫\), \(0\), \(∞\)]\(
\*FractionBox[\(π\), \(2\ x\)] \[DifferentialD]x\)\) *)
in Latex:
$$\int_0^{\infty } \frac{\pi }{2 x} \, dx$$
There is no warning that the result might be infinite or not existent.
Applying Simplify[]
or FullSimplify[]
to $f$ does not help.
Numerically, however, the divergence is revealed
f // N
During evaluation of In[1418]:= NIntegrate::slwcon: Numerical integration converging too slowly; suspect one of the following: singularity, value of the integration is 0, highly oscillatory integrand, or WorkingPrecision too small. >>
and further error Messages which I have dropped here.
(* Out[1418]= 366.404 *)
I suggest to consider this behaviour a bug.
Taking finite integration regions reveals the logarithmic divergnce of the double integral:
fi = Integrate[ 1/(1 + x^2 y^2), {x, 0, t}, {y, 0, t}, Assumptions -> t > 0]
(* Out[1411]= 1/2 I (PolyLog[2, -I t^2] - PolyLog[2, I t^2]) *)
Series[fi, {t, ∞, 2}] // Normal
(* Out[1416]= 1/t^2 - π Log[1/t] *)
Limit[fi, t -> ∞]
(* Out[1400]= ∞ *)
Integral[1/x,{x,0,Infinity]
? $\endgroup$