# Thread: Hard Integral

1. ## Hard Integral

Find the following integral:

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

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

2. 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

3. 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:

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

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

= $\sum^{\infty}_{n=0} \frac{-x^{n+0.5}}{(n+0.5)n!} +C$

4. 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

5. Originally Posted by Jhevon
is the integral indefinite? if so, there is no solution in elementary functions, you would need the erf function
..

6. 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...

7. 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

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