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?
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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 $\displaystyle \int_0^{\infty} e^{-x} \sqrt{1+e^{-2x}}\, dx \le \int_0^{\infty} \sqrt{2} \, e^{-x} dx. $
Ah yes, and clearly the larger integral converges, and so alas! Thanks
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