Polar coordinate question

∫ ∫ xy/sqrt (x^2+y^2) dxdy

Limits of integration are both from 1 to 2.

Converting this directly to polar coordinates is painful!

Evaluating this directly:
∫(y = 1 to 2) ∫(x = 1 to 2) [xy / √(x^2+y^2)] dx dy
= ∫(y = 1 to 2) y√(x^2+y^2) {for x = 1 to 2} dy
= ∫(y = 1 to 2) [y√(y^2 + 4) - y√(y^2 + 1)] dy
= [(1/3) (y^2 + 4)^(3/2) - (1/3) (y^2 + 1)^(3/2)] {for y = 1 to 2}
= (1/3) [(y^2 + 4)^(3/2) - (y^2 + 1)^(3/2)] {for y = 1 to 2}
= (1/3) {[8^(3/2) - 5^(3/2)] - [5^(3/2) - 2^(3/2)]}
= (1/3) {[8 * 2^(3/2) - 5^(3/2)] - [5^(3/2) - 2^(3/2)]}, since 4^(3/2) = 8
= (1/3) [9 * 2^(3/2) - 2 * 5^(3/2)].

I hope this helps!