Two hard problems of improper integrals

Question

I have 2 problems that I have been stuck on: Check the absolute convergence and convergence of these improper integrals:
a) $\int\limits_0^\infty x^p\sin(x^q)dx$ $(q\neq 0)$.
b) $\int\limits_0^\infty\dfrac{x^p\sin x}{1+x^q}dx$ $(q\geq 0)$.
This is all I have done:
b) I split the the improper integral at $1$. It's easy to notice that $\dfrac{x^p\sin x}{1+x^q}$~$\dfrac{x^{p+1}}{1+x^q}$ when $x\to 0$, so $\int\limits_0^1\dfrac{x^p\sin x}{1+x^q}dx$ converges iff $q-p-1<1$.
Any further help I would be very much appreciate.