Hallo,
für \( 0 < \varepsilon < 1 \) gilt:
\( \int \limits_{\varepsilon}^{1} \frac{1}{\sqrt{x}} \cdot e^{ -\sqrt{x}} \,dx \overset{y = \sqrt{x},\, dy = \frac{1}{2\sqrt{x}}dx}{=} 2 \int \limits_{\sqrt{\varepsilon}}^{1} e^{ -y} \,dy = 2 \cdot (e^{-\sqrt{\varepsilon}} - \frac{1}{e}) = 2e^{-\sqrt{\varepsilon}} - \frac{2}{e} \)
\( \Longrightarrow \int \limits_{0}^{1} \frac{1}{\sqrt{x}} \cdot e^{ -\sqrt{x}} \,dx = \lim\limits_{\varepsilon\to0^+}\int \limits_{\varepsilon}^{1} \frac{1}{\sqrt{x}} \cdot e^{ -\sqrt{x}} \,dx = \lim\limits_{\varepsilon\to0^+}(2e^{-\sqrt{\varepsilon}} - \frac{2}{e}) = 2 - \frac{2}{e} \)
