does this integral converge?

**szpengchao** · April 8th 2008, 07:59 AM

integral( x^p * exp(-x^q)dx, 0,infinite) where, p,q >0

**Oli** · April 8th 2008, 09:01 AM

Yes it does. Can't remember why though.

**PaulRS** · April 8th 2008, 10:08 AM

Yep, it converges.

$\int_0^{\infty}x^p\cdot{e^{-x}}dx=\Gamma(p+1)$

To see why it converges, compute the limit $\lim_{x\rightarrow{+\infty}}\frac{x^p\cdot{e^{-x^{q}}}}{\frac{1}{x^2}}$
What can you say about $\int_1^{\infty}{\frac{dx}{x^2}}$ ? and then what happens with $\int_1^{\infty}{x^p\cdot{e^{-x^{q}}}dx}$ ?

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