Show that $f$ is integrable with respect to the Lebesgue measure

Question

Let $\alpha \gt 0 \forall \alpha \in\mathbb{R}$, $f_{\alpha}:]0,\infty[\to\mathbb{R}$, $x\mapsto e^{-\alpha x}\left(\frac{sin(x)}{x}\right)^3$. The goal ist to show that $f_{\alpha}$ is integrable with respect to the Lebesgue-measure, which coincides with the Borel measure in this case. I already showed that $f_{\alpha}$ is measurable and know that the condition for $f_{\alpha}$ being integrable is that $$\int_X\lvert{f(x)}\rvert\mathrm{d}\lambda\lt\infty$$ with $X=]0,\infty[$. Since I am fairly new to this topic, I don't know where to start. I struggle to write out the integral since it can attain negative values and lack a general strategy for exercises like this. Any help is greatly appreciated!

Answer 1

$\frac {\sin\, x} x$ is abounded function on $(0,\infty)$. If $|\frac {\sin\, x} x| \leq M$ then $\int_0^{\infty} |f_{\alpha} (x)| \, dx \leq M^{3}\int_0^{\infty} e^{-\alpha x}\, dx =\frac {M^{3}} {\alpha}$. [ Boundedness of $\frac {\sin\, x} x$ is a consequence of the following facts: this function is continuous, it approaches $1$ as $ x\to 0$ and approaches $0$ as $x \to \infty$].