$A.$ $I_{1} < I_{2} < I_{3}$. $B.$ $I_{3} < I_{2} < I_{1}$. $C.$ $I_{2} < I_{3} < I_{1}$. $D.$ $I_{2} < I_{1} < I_{3}$.

D

#### 解析：

$\mathrm{I}_{3}=\int_{0}^{3 \pi} \mathrm{e}^{x^{2}} \sin x \mathrm{~d} x=\int_{0}^{2 \pi} \mathrm{e}^{x^{2}} \sin x \mathrm{~d} x+\int_{2 \pi}^{3 \pi} \mathrm{e}^{x^{2}} \sin x \mathrm{~d} x=\mathrm{I}_{2}+\int_{2 \pi}^{3 \pi} \mathrm{e}^{x^{2}} \sin x \mathrm{~d} x$

$I_{3}=\int_{0}^{3 \pi} \mathrm{e}^{x^{2}} \sin x \mathrm{~d} x=\int_{0}^{\pi} \mathrm{e}^{x^{2}} \sin x \mathrm{~d} x+\int_{\pi}^{3 \pi} \mathrm{e}^{x^{2}} \sin x \mathrm{~d} x=\mathrm{I}_{1}+\int_{\pi}^{3 \pi} \mathrm{e}^{x^{2}} \sin x \mathrm{~d} x$
$\int_{\pi}^{3 \pi} \mathrm{e}^{x^{2}} \sin x \mathrm{~d} x=\int_{\pi}^{2 \pi} \mathrm{e}^{x^{2}} \sin x \mathrm{~d} x+\int_{2 \pi}^{3 \pi} \mathrm{e}^{x^{2}} \sin x \mathrm{~d} x$
$=\int_{\pi}^{2 \pi} \mathrm{e}^{x^{2}} \sin x \mathrm{~d} x+\int_{\pi}^{2 \pi} \mathrm{e}^{(t+\pi)^{2}} \sin (t+\pi) \mathrm{d}(t+\pi)$
$=\int_{\pi}^{2 \pi} \mathrm{e}^{x^{2}} \sin x \mathrm{~d} x-\int_{\pi}^{2 \pi} \mathrm{e}^{(x+\pi)^{2}} \sin x \mathrm{~d} x=\int_{\pi}^{2 \pi}\left[\mathrm{e}^{x^{2}}-\mathrm{e}^{(x+\pi)^{2}}\right] \sin x \mathrm{~d} x > 0$
$\therefore \mathrm{I}_{3} > \mathrm{I}_{1}$