Two Indefinite Integrals Looking for some hints to evaluate the following integrals (with complex analysis or otherwise):
$$\int_0^\infty\frac{x^{p-1}}{x+1}\,dx,\;\;\;\; 0<p<1,$$
$$\int_{-\infty}^\infty e^{-s^2+isz}\,ds,\;\;\;\;z\in\mathbb{C}.$$
Thanks, I'm pretty stuck on both.

First rewrite it as
$$ \displaystyle \int_{0}^{\infty}\dfrac{1}{{(x+1)x^{1-p}}}$$
If $x\in (0,1)$ then $x^{2-p}+x^{1-p}>x^{1-p}$. Therefore 
$$ \displaystyle \int_{0}^{1}\dfrac{1}{{(x+1)x^{1-p}}}dx<\int_{0}^{1}\dfrac{1}{{x^{1-p}}}dx$$
Since the larger one converges by $p$ test the smaller one converges by comparison test.
If $x\in (1,\infty)$ then $x^{2-p}+x^{1-p}>x^{2-p}$. Therefore 
$$ \displaystyle \int_{1}^{\infty}\dfrac{1}{{(x+1)x^{2-p}}}dx<\int_{1}^{\infty}\dfrac{1}{{x^{2-p}}}dx$$
Since the larger one converges by $p$ test the smaller one converges by comparison test.
