Question

determine if internal converges or diverges: integral (0-inf.) of e^-sqrt(x)/sqrt(x)

Answer 1

integral*

Answer 2

No. \[\int_0^{\infty}\frac{e^{-\sqrt{x}}}{\sqrt{x}}dx=\int_0^1\frac{e^{-\sqrt{x}}}{\sqrt{x}}dx+\int_1^{\infty}\frac{e^{-\sqrt{x}}}{\sqrt{x}}dx \]The first integral diverges. Not sure about the second one, but that doesn't really matter.

Answer 3

Since \( e^{-\sqrt{x}}\) is monotone decreasing on (0,∞), its minimum value on (0,1] is at 1, and is equal to \(1/e\).