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

by Michael Maier   Last Updated January 11, 2019 12:20 PM - source

let $\alpha \gt 0 \forall \alpha \in\mathbb{R}$, $f_{\alpha}:]0,\infty[\to\mathbb{R}$, $x\to e^{-\alpha x}\left(\frac{sin(x)}{x}\right)^3$. 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 $$\int_X\abs{f(x)}d\lambda\lt\infty$$ 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!

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