Answer
$9(e-1)$
Work Step by Step
Write the region $R$ of integration.
$R=${$ (x,y) | 0 \leq x \leq 1/y, 1 \leq y \leq 10$}
$I=\int_{1}^{10} \int_{0}^{1/y} ye^{xy} \ dx \ dy \\=\int_{1}^{10} [e^{(xy)}]_{0}^{1/y} \ dy \\=\int_{1}^{10} (e-1) \ dy \\=[(e-1) \times y]_{1}^{10} \\=10(e-1)-(e-1 ) \\=9(e-1)$
