Parametric integrals

Question

For $0 \le \alpha \lt \infty$ and $p \gt 0$ evaluate integral $$I(\alpha, p) = \int_1^\infty e^{-\alpha x} \frac{\cos x}{x^p} \, dx$$ This should be solved by using theorem about differentiation by parameter. I've tried with both parameters $p$ and $\alpha$. In both cases I've managed to prove that $\frac{\partial}{\partial \alpha} f(x, \alpha, p)$ and $\frac \partial {\partial p} f(x, \alpha, p)$ are continuous by $(x, \alpha, p) \in [1, \infty) \times [0, \infty) \times (0, \infty)$, where $f(x, \alpha, p) = e^{-\alpha x} \frac{\cos x}{x^p}$. Also, I've manged to prove convergence of $\int_1^\infty e^{-\alpha x} \frac{\cos x}{x^p} \, dx$ and uniform convergence by both $\alpha$ and $p$ of $\int_1^\infty \frac \partial {\partial \alpha} f(x, \alpha, p) \,dx$ and $\int_1^\infty \frac \partial {\partial p} f(x, \alpha, p) \, dx$ respectively (although I could not prove convergence of $\int_1^\infty \frac \partial {\partial \alpha} \, ex$ for $p \in (0, 1)$). Now, I should evaluate one of those integrals, but it was too hard for me. Any help or advice is appreciated, and solution particularly.