$\int_{0}^{\infty}{dx \over (1+x)x^\alpha} = {\pi \over \sin(\pi (1-\alpha))} $ Can you help me show that 
$$\int_{0}^{\infty}{dx \over (1+x)x^\alpha} = {\pi \over \sin(\pi (1-\alpha))}$$ 
such that $\alpha \lt1$?
 A: The integral 
\begin{equation}\int_{0}^{+\infty}\frac{x^{-\alpha}}{x+1}\, dx\end{equation}
converges if $0<\alpha<1$. Take for example $a=-1$. Then,
\begin{equation}\int_{0}^{+\infty}\frac{x}{x+1}\, dx=\int_{0}^{+\infty}1-\frac{1}{x+1}\, dx=\lim_{x\to +\infty}x-\ln(x+1)=+\infty\end{equation}
We shall calculate that integral (for $0<\alpha<1$) with complex analysis:
The associated complex function is
\begin{equation}f(z)=\frac{z^{-\alpha}}{z+1}\end{equation}
where $z^{-\alpha}$ is chosen so that it is continuous and holomorphic in $\mathbb{C}\setminus\left\{(x,0):x\ge 0\right\}$. We shall integrate over the keyhole integral:
We have 
\begin{equation}\int_{C}f(z)\, dz=\int_{\gamma_ \epsilon}f(z)\, dz+\int_{[\epsilon i,R+\epsilon i]}f(z)\, dz+\int_{\gamma_R}f(z)\, dz+\int_{[ R-\epsilon i,-\epsilon i]}f(z)\, dz\end{equation}
The main observation here is that for $x>0$,
\begin{gather}\lim_{\epsilon\to 0^+}\log(x+\epsilon i)=\lim_{\epsilon\to 0^+}\log\sqrt{x^2+\epsilon^2}+i\arg(x+\epsilon i)=\log x\notag\\
\lim_{\epsilon\to 0^+}\log(x-\epsilon i)=\lim_{\epsilon\to 0^+}\log\sqrt{x^2+\epsilon^2}+i\arg(x-\epsilon i)=\log x+2\pi i
\end{gather}
Therefore,
\begin{equation}\int_{[\epsilon i,R+\epsilon i]}f(z)\, dz=\int_{0}^{R}f(x+\epsilon i)\, dx=\int_{0}^{R}\frac{x^{-\alpha}}{x+\epsilon i+1}\, dx=
\int_{0}^{R}\frac{e^{\log(x+\epsilon i)(-\alpha)}}{x+\epsilon i+1}\, dx\to\int_{0}^{R}\frac{e^{\log x(-\alpha)}}{x+1}\, dx= \int_{0}^{R}\frac{x^{-\alpha}}{x+1}\, dx
\end{equation}
as $\epsilon\to 0^+$ while
\begin{equation}\int_{[R-\epsilon i,-\epsilon i]}f(z)\, dz=\int_{R}^{0}f(x-\epsilon i)\, dx=-\int_{0}^{R}\frac{e^{\log(x-\epsilon i)(-\alpha)}}{x-\epsilon i+1}\, dx\to-\int_{0}^{R}\frac{e^{[\log x+2\pi i](-\alpha)}}{x+1}\, dx= -e^{-2\pi i\alpha}\int_{0}^{R}\frac{x^{-\alpha}}{x+1}\, dx
\end{equation}
again as $\epsilon\to 0^+$.
By the Residue theorem,
\begin{equation}\int_{C}f(z)\, dz=2\pi Res_{-1}(f)=(-1)^{-\alpha}=2\pi e^{-(\log-1)\alpha}=2\pi ie^{-i\pi\alpha}\end{equation}
The integrals of $f$ over $\gamma_R$ and $\gamma_{\epsilon}$ converge to $0$ as $R\to +\infty$, $\epsilon\to 0^+$ (why?)
and so
\begin{equation}2\pi i e^{-i\pi\alpha}=(1-e^{-2\pi i\alpha})\int_{0}^{+\infty}\frac{x^{-\alpha}}{x+1}\, dx\end{equation}
which implies
\begin{equation}\int_{0}^{+\infty}\frac{x^{-\alpha}}{x+1}\, dx=\frac{\pi}{\sin \pi \alpha}={\pi \over \sin(\pi (1-\alpha))}\end{equation}
