Original integral I = ∫∫

2. The integral region is symmetrical about x, and cos(xy) is an even function of y;

The integral region is symmetrical about y axis, and cos(xy) is an even function of X. 。

Let D 1 be a quarter circle of the first quadrant, then

Original integral I = 4 ∫∫

= 4∫& lt； 0，π/2 & gt； dt∫& lt； 0，r & gt[e^(R^2)cos(R^2*sintcost)]RdR，

The limitations sought are

lim & ltr→0 & gt； 4∫& lt； 0，π/2 & gt； dt∫& lt； 0, r> [e (R2) cos (R2 * sintcost)] rdr/(π R2) (type 0/0)

= lim & ltr→0 & gt； 4∫& lt； 0，π/2 & gt； dt[e^(r^2)cos(r^2*sintcost)]r/(2πr)

= 4∫& lt； 0，π/2 & gt； dt[ 1/(2π)]= 1。