# Integral of x^(m-1)/(1+x) wrt x?

## Main Question or Discussion Point

How does this:
$\int_0^\infty\frac{x^{m-1}}{1+x} dx$
equal
$\frac{\pi}{sin(m\pi)}$
?
It has been simply stated as a fact in a proof of the so called "Euler reflection formula" in a textbook.
I have tried the usual ways, substitution, integration by parts and even series expansion of 1/1+x but I can't find how the above equality is true.

maajdl
Gold Member
I suggest you to first try on some special cases.
For example: m=1/2, m=3/2, m=1/4.
Then, also, analyze the domain of convergence of this integral.
Besides, I think there is no elementary primitive.
Have a look on the wiki article: http://en.wikipedia.org/wiki/Reflection_formula .
Maybe you might prove the relation of your integral to the product $\Gamma$(m)$\Gamma$(1-m)

maajdl
Gold Member
Far from obvious!
See pages 13-17

http://warwickmaths.org/files/gamma.pdf [Broken]

Last edited by a moderator:
Thanks. So I guess I will just have to accept that as a fact. No easy and 'real' way to obtain the result without resorting to other forms of Gamma function.

maajdl
Gold Member