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Test the improper integral ∫ e-x/2 for convergence

by RoRi

Test the following improper integral for convergence:

    \[ \int_0^{\infty} \frac{1}{\sqrt{e^x}} \, dx. \]

The integral converges.

Proof. We compute directly,

    \begin{align*} \int_0^{\infty} \frac{1}{\sqrt{e^x}} \, dx &= \int_0^{\infty} e^{-\frac{x}{2}} \, dx \\[9pt] &= \lim_{b \to +\infty} \left( \int_0^b e^{-\frac{x}{2}} \, dx \right) \\[9pt] &= \lim_{b \to +\infty} \left( -2 e^{-\frac{x}{2}} \Bigr \rvert_0^b \right) \\[9pt] &= \lim_{b \to +\infty} \left( -2 e^{-\frac{b}{2}} + 2 \right) \\[9pt] &= 2. \qquad \blacksquare \end{align*}

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