Hard Integral

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April 17th 2008, 10:47 PM
Ideasman

Hard Integral

Find the following integral:

$\int{\frac{e^{-x}}{\sqrt{x}}~dx}$

Unable to use integration by parts I believe for this problem..
April 17th 2008, 11:08 PM
Jhevon

Quote:

Originally Posted by Ideasman

Find the following integral:

$\int{\frac{e^{-x}}{\sqrt{x}}~dx}$

Unable to use integration by parts I believe for this problem..

is the integral indefinite? if so, there is no solution in elementary functions, you would need the erf function
April 17th 2008, 11:22 PM
lllll

not sure if this will be of any help, but you could also express [tex] e^{-x} as an infinite series then integrate it out.

so you would have:

$\int \frac{\sum^{\infty}_{n=0} \frac{-x^n}{n!}}{\sqrt{x}} dx$

= $\int \sum^{\infty}_{n=0} \frac{-x^{n-0.5}}{n!} dx$

= $\sum^{\infty}_{n=0} \frac{-x^{n+0.5}}{(n+0.5)n!} +C$
April 18th 2008, 04:17 AM
topsquark

Quote:

Originally Posted by Ideasman

Find the following integral:

$\int{\frac{e^{-x}}{\sqrt{x}}~dx}$

Unable to use integration by parts I believe for this problem..

If this is supposed to be
$\int_0^{\infty} {\frac{e^{-x}}{\sqrt{x}}~dx}$
then this is a gamma function.

-Dan
April 18th 2008, 04:21 AM
mr fantastic

1 Attachment(s)

Quote:

Originally Posted by Jhevon

is the integral indefinite? if so, there is no solution in elementary functions, you would need the erf function

..
April 19th 2008, 06:44 PM
Jhevon

Quote:

Originally Posted by topsquark

If this is supposed to be
$\int_0^{\infty} {\frac{e^{-x}}{\sqrt{x}}~dx}$
then this is a gamma function.

-Dan

yes, if we had limits like that, then a substitution would turn the integrand into $2e^{-u^2}$ which is fun to do...
April 19th 2008, 06:54 PM
PaulRS

Quote:

Originally Posted by topsquark

If this is supposed to be
$\int_0^{\infty} {\frac{e^{-x}}{\sqrt{x}}~dx}$
then this is a gamma function.

-Dan

Yes

Let $u=\sqrt[]{x}$ then $\int_0^{\infty} {\frac{e^{-x}}{\sqrt{x}}dx}=2\int_0^{\infty} {e^{-u^2}du}$ $=\int_{-\infty}^{\infty} {e^{-u^2}du}=\sqrt[]{\pi}$

Gaussian integral - Wikipedia, the free encyclopedia
April 19th 2008, 06:58 PM
Krizalid

The well known gaussian integral. There're lots of proofs of its result. :D

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