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.

For the first integral, you can set $t=\dfrac{x}{x+1}$ so that $\mathrm{d}x=\dfrac{\mathrm{d}t}{(1-t)^2}$, and you have

 $$\int_0^\infty \frac{x^{p-1}}{x+1}\,\mathrm{d}x=\int_0^1t^{p-1}(1-t)^{-p}\,\mathrm{d}t=\mathrm{B}(p,1-p)$$
 where $\mathrm{B}(a,b)$ denotes the Beta function.

For the second integral, consider completing the square, then recall that
$$\int_{-\infty}^\infty e^{-x^2}\,\mathrm{d}x=\sqrt\pi$$
