For $x \in \mathbb{R}^{+}$, we define $$f(x) = \int_{0}^{\infty} \frac{1 - \cos t}{t^{2}} \mathrm{e}^{-xt} \mathrm{~d}t$$ Show that $f$ is defined and continuous on $[0, +\infty[$ and of class $C^{2}$ on $]0, +\infty[$.
For $x \in \mathbb{R}^{+}$, we define
$$f(x) = \int_{0}^{\infty} \frac{1 - \cos t}{t^{2}} \mathrm{e}^{-xt} \mathrm{~d}t$$
Show that $f$ is defined and continuous on $[0, +\infty[$ and of class $C^{2}$ on $]0, +\infty[$.