证明: (1) 曲线积分

$
\int_C \mathrm{e}^{-2 x y} \cos \left(x^2-y^2\right) \mathrm{d} x-\mathrm{e}^{-2 x y} \sin \left(x^2-y^2\right) \mathrm{d} y
$

与路径无关;
(2) 证明: $\lim _{R \rightarrow+\infty}\left(\int_0^R \cos x^2 \mathrm{~d} x-\int_0^R \mathrm{e}^{-2 x^2} \mathrm{~d} x\right)=0$;
(3) 证明: $\lim _{R \rightarrow+\infty}\left(\int_0^R \sin x^2 \mathrm{~d} x-\int_0^R \mathrm{e}^{-2 x^2} \mathrm{~d} x\right)=0$.