calculus – Does double integral require reversal to solve?

I’m having difficulty solving the following double integral without reversing it. Is that the best approach? I have the sense that this problem is much simpler than I’m making it:

$${large int_{1}^{2}int_{1}^{x^3}xcos{y}dydx}$$

Without reversing the order of the integrals, I get:

$${large int_{1}^{2}[xsin{x^3} – xsin{1}]dx }$$

Better seems to be a reversal of integral order:

$${large int_{1}^{8}int_{sqrt[3]{y}}^{2}xcos{y}dxdy}$$

Have I reversed the order correctly? And does the answer reduce to $sin{8} – sin{1}$ ?