Use simulation with antithetic variables and find

$$\int_{-\infty}^{\infty} \int_0^\infty\, sin(x+y)e^{-x^2+4x-y}\,dx\,dy,$$
so, my question and doubt is how struggle with the infinite limit ?

It is easy to me this problem $\int_0^1e^x\,dx$ or similar, but do not know how the previous.

Help me please.