Some problem with Euler's Reflection Formula?

Question

$$\Gamma(z)\Gamma(1-z)=\int_0^{\infty}x^{z-1}e^{-x}dx\int_0^{\infty}x^{-z}e^{-x}dx=\int_0^{\infty}(2x)^{-1}e^{-2x}(2dx)=\Gamma(0)\to\infty??$$

Answer 1

I don't think your operation is valid. Compare:

$$ \int_a^b x \; dx \int_a^b 1/x \; dx = \frac12(b^2-a^2)(\ln b - \ln a) \neq \int_a^b 1 dx =b-a $$