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

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

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*}

Point out an error, ask a question, offer an alternative solution (to use Latex type [latexpage] at the top of your comment):