Show $\int_{M}^{\infty}\left|\frac{e^{-x^a}\cos(Cx)}{x^{b+1}}\right|dx<\frac{1}{aM^{a+b}e^{M^a}}$?

Question

is there a way to show the inequality

\begin{equation}\int_{M}^{\infty}\left|\frac{e^{-x^a}\cos(Cx)}{x^{b+1}}\right|dx<\frac{1}{aM^{a+b}e^{M^a}}\end{equation}

for positive constants $M$ and $C$ and for $ 0<a<2$ and $0<b<a$? It could also be, that the approximation does not hold, I wasn't able to show either one yet, though.

Thanks in advance!

Answer 1

$$\int_{M}^{\infty}\left|\frac{e^{-x^a}\cos(Cx)}{x^{b+1}}\right|dx\le \int_{M}^{\infty}ax^{a-1}\frac{e^{-x^a}}{ax^{a-1}x^{b+1}}dx\le \int_{M}^{\infty}ax^{a-1}\frac{e^{-x^a}}{aM^{a+b}}=\frac{1}{aM^{a+b}e^{M^a}}$$