Show that the function \(\mathbb{R} i x \mapsto \exp \left(-x^{\alpha}ight)\) is \(\lambda^{1}(d x)\)-integrable over the set \([0, \infty)\) for every \(\alpha>0\).

[find integrable majorants \(u\) resp. \(w\) if \(0 \leqslant x \leqslant 1\) resp. \(1