# Math Help - Convergence of an integral

1. ## Convergence of an integral

How could you tell that the improper integral from 0 to infinity of:

e^(-x)*sqrt(1+e^(-2x)) dx converges WITHOUT actually computing it?

It is obvious from the graph, but how can you argue this otherwise?

2. Originally Posted by zhupolongjoe
How could you tell that the improper integral from 0 to infinity of:

e^(-x)*sqrt(1+e^(-2x)) dx converges WITHOUT actually computing it?

It is obvious from the graph, but how can you argue this otherwise?
Consider

$
\int_0^{\infty} e^{-x} \sqrt{1+e^{-2x}}\, dx \le \int_0^{\infty} \sqrt{2} \, e^{-x} dx.
$

3. Ah yes, and clearly the larger integral converges, and so alas! Thanks