# Strange results with a double definite integral

Doing Integrate[1/(x y),{x,0,1},{y,0,1}] returns zero, although the integral is clearly divergent; NIntegrate attempts to compute it, then gives up at 2.15434*10^9 with some error messages.

Is there a way to get a correct answer?

There are several complaints here about strange evaluations of integrals, but none of them are quite like this. The closest one, Strange result for divergent double integral $\int _0^{\infty }\int _0^{\infty }\frac{1}{x^2 y^2+1}dydx$ does not have any answers, so I decided to post this as another instance of the same problem.

• What MMA version and operating system are you on? I get the same on 10.0.2.0 on macOS 10.12.4 – Marius Ladegård Meyer May 6 '17 at 10:09
• @MariusLadegårdMeyer Mine is 11.0.1.0 on WIndows 10 – მამუკა ჯიბლაძე May 6 '17 at 10:48
• Version 10.0 gives the error: "Integral of 1/x does not converge on {0,1}." – mattiav27 May 6 '17 at 11:22
• @მამუკაჯიბლაძე I meant that using your function, Mathematica gives correctly an error. I meant it works with my version of Mathematica. – mattiav27 May 6 '17 at 12:23
• I will file a bug report but I should mention that any "simple" fix might turn out to break things. – Daniel Lichtblau May 8 '17 at 21:11

This has been fixed as of version 11.3.0:

Integrate[1/(x y), {x, 0, 1}, {y, 0, 1}] // Head


Integrate::idiv: Integral of 1/(x y) does not converge on {0, 1}.

(* Integrate *)


The limit of (Integrate[1/(x y), {x, a, 1}, {y, a, 1} ], a -> 0) is Infinity.

In:

Clear[n]
f[a_] := Integrate[1/(x y), {x, a, 1}, {y, a, 1} ]
Limit[f[a], a -> 0]
Table[{10^(-n), f[10^(-n)] // N}, {n, 200, 100, -1}] //
ListLinePlot[#, PlotRange -> Full] &

• As a matter of fact, Integrate[1/(x y), {x, a, 1}, {y, a, 1} ] is Log[a]^2 (under some restrictions on a) – მამუკა ჯიბლაძე May 7 '17 at 5:28