Integrate and find closed form for $\int_0^\infty\frac{\sin x^n}{n\pi}dx$

I’m working on a practice exam for my analysis class, and I was asked to find a general form for$$\int_0^\infty\frac{\sin x^n}{n\pi}dx$$When I first looked at this, my mind instantly went to the Dirchlet Integral. Its been some time and I haven’t been able to see the relation.