Find the following integral:
$\displaystyle \int{\frac{e^{-x}}{\sqrt{x}}~dx}$
Unable to use integration by parts I believe for this problem..
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:
$\displaystyle \int \frac{\sum^{\infty}_{n=0} \frac{-x^n}{n!}}{\sqrt{x}} dx$
=$\displaystyle \int \sum^{\infty}_{n=0} \frac{-x^{n-0.5}}{n!} dx$
=$\displaystyle \sum^{\infty}_{n=0} \frac{-x^{n+0.5}}{(n+0.5)n!} +C $
If this is supposed to be
$\displaystyle \int_0^{\infty} {\frac{e^{-x}}{\sqrt{x}}~dx}$
then this is a gamma function.
-Dan
Yes
Let $\displaystyle u=\sqrt[]{x}$ then $\displaystyle \int_0^{\infty} {\frac{e^{-x}}{\sqrt{x}}dx}=2\int_0^{\infty} {e^{-u^2}du}$$\displaystyle =\int_{-\infty}^{\infty} {e^{-u^2}du}=\sqrt[]{\pi}$
Gaussian integral - Wikipedia, the free encyclopedia