hier mal meine Rechnung:
a)
$$ \phi(x)=\int_{0}^{\infty}du e^{-ux}=\frac{1}{x}\\\frac{d}{dx}\phi(x)=\int_{0}^{\infty}du (\frac{d}{dx}e^{-ux})=-\int_{0}^{\infty}du*ue^{-ux}=-\frac{1}{x^2}\\\frac{d^n}{dx^n}\phi(x)=\int_{0}^{\infty}du (\frac{d^n}{dx^n}e^{-ux})=(-1)^n\int_{0}^{\infty}du*u^ne^{-ux}=(-1)^nn!\frac{1}{x^{n+1}}\\\to \int_{0}^{\infty}du*u^ne^{-ux}=n!\frac{1}{x^{n+1}}\\\to x=1:\int_{0}^{\infty}du*u^ne^{-u}=n! $$
b)
$$ \int_{0}^{1}dxx^p(ln(x))^m =\int_{0}^{1}dxe^{pln(x)}(ln(x))^m\\\text{Substituiere }\\z=ln(x)\\\frac{dz}{dx}=1/x\\dx=dzx=dze^z\\\int_{0}^{1}dxe^{pln(x)}(ln(x))^m=\int_{-\infty}^{0}dze^ze^{pz}z^m\\=\int_{-\infty}^{0}dze^{(p+1)z}z^m=-\int_{0}^{-\infty}dze^{(p+1)z}z^m\\\text{Substituiere }\\z=-k\\dz=-dk\\-\int_{0}^{-\infty}dze^{(p+1)z}z^m\\=(-1)^m\int_{0}^{\infty}dke^{-(p+1)k}k^m\\\text{Substituiere}\\(p+1)k=k'\\dk*(p+1)=dk'\\(-1)^m\int_{0}^{\infty}dke^{-(p+1)k}k^m\\=(-1)^m/(p+1)^{m+1}\int_{0}^{\infty}dk' e^{-k'}k'^m=\frac{(-1)^mm!}{(p+1)^{m+1}} $$
Für p<=-1 konvergiert das Integral nicht.
