integral converge or diverge

**bmixon** · July 23rd 2009, 06:17 PM

does this integral converge or diverge?

a) int from 0 to infinity of e^(-x)/sqrroot(x)

b) int from o to pi/2 of 1/sqroot(sec(x)-1))

**Krizalid** · July 23rd 2009, 06:29 PM

For the first one, split it into two pieces: $\int_{0}^{\infty }{\frac{dx}{e^{x}\sqrt{x}}}=\int_{0}^{1}{\frac{dx} {e^{x}\sqrt{x}}}+\int_{1}^{\infty }{\frac{dx}{e^{x}\sqrt{x}}}.$

The first one converges by simple limit comparison test with $\int_0^1\frac{dx}{\sqrt x},$ the second one just require a simple bound to determine its convergence. I'll let you to figure out.

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