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 (r2) / ln(r2)] r dr(dtheta)
2 -- 2/sin(theta)

B=
pi/2 -------∞
∫---------- ∫ [sin (r2) / ln(r2)] 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!:)

To quote Ben:

Please use Tex!

The integral of r/ln(r) diverges as r->inf. Since sin(r^2) oscillates,but doesn't go to zero, I suspect the integral diverges.

deleted nonsense