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

[s0ft]

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]

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]

Far from obvious!
See pages 13-17

http://warwickmaths.org/files/gamma.pdf

 

[s0ft]

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]

See also:

http://www.proofwiki.org/wiki/Euler's_Reflection_Formula

http://mathoverflow.net/questions/76399/one-line-proof-of-the-eulers-reflection-formula

http://math.stackexchange.com/questions/578609/euler-reflection-formula-via-wielandt-theorem

http://books.google.co.uk/books?id=...#v=onepage&q=Euler reflection formula&f=false

From the last reference, it appears that maybe some contour integration in the complex domain might be helpful. (p10)