Answer
$\dfrac{2}{\pi}$
Work Step by Step
Re-arrange the given integral as follows:
$\int_{0}^{\pi} \int_{0}^{1} (y \cos xy) dy dx=\int_{0}^{1}y (\dfrac{\sin xy}{y})]_{0}^{\pi} dy$
This implies that
$\int_{0}^{1}y (\dfrac{\sin xy}{y})]_{0}^{\pi} dy=\int_{0}^{1} \sin (\pi y) dy$
Hence, $[-\dfrac{cos xy}{\pi}]_0^1=\dfrac{2}{\pi}$
