Let $x,y \in \mathbb{C}, \Re(x)>0, \Re(y)>0$.
How can I show that $t \mapsto \frac{t^{x-1}}{1+t^y}$ is Lebesgue integrable on $(0,1)$?
And furthermore I want to show $$\int_{(0,1)}\frac{t^{x-1}}{1+t^y}d\lambda(t)=\sum_{n=0}^{\infty}\frac{(-1)^n}{x+ny}$$ where $t^x=\exp(x\ln(t)) , t^y=\exp(y\ln(t))$.
I know how to show
$$\int_0^1\frac{x^{p-1}}{1+x^q}dx=\sum_{n=0}^{\infty}\frac{(-1)^n}{qn+p}$$ for $p,q>0$ but does that help me here?
The hint was to write $\frac{t^{x-1}}{1+t^y}$ as $\sum_{n=0}^{\infty}(-1)^nt^{x-1+ny}$ and this series converges. And $2 \Re t^y \ge t^{\Re y}$ for $t$ close to $1.$