Improper Multiple Integrals (2)

[kingwinner]

I find this to be a very tough problem:

1) Determine whether the improper integral I
∞ ∞
∫ ∫ [sin (x² + y²) / ln(x² + y²)] dxdy
2 2
converges or diverges.

All I can think of and try is by changing it to polar coordinates:

I=A+B where

A=
pi/4--- ∞
∫ ----- ∫ [sin (r^2) / ln(r^2)] r dr(dtheta)
2 -- 2/sin(theta)

B=
pi/2 -------∞
∫---------- ∫ [sin (r^2) / ln(r^2)] r dr(dtheta)
pi/4 ---2/cos(theta)


But how can I show that each of them converges (or diverges)?

Can someone please help me out? Thank you!

 

[Dick]

What can you say about r*sin(r^2)/ln(r^2)? I would say it is oscillatory and tending in amplitude to infinity as r->infinity. You would need a very liberal version of the definition of 'integral' to be able to say that that exists. What's your definition of integral? Can't you find anything in that definition that would allow to say "This diverges."? I wouldn't even try to actually compute an iterated integral, the function form fits polar coordinates but the boundaries don't.