In the Riemann's paper "On the Number of Primes Less Than a Given Magnitude" we have $\zeta(s)$ defined for $\Re(s)>1$ by this integral:
$$\tag{1}\zeta(s)=\frac{1}{\Gamma(s)} \int_0^{\infty } \frac{x^{s-1}}{e^x-1} \, dx$$
Then he considers the following integral:
$$\tag{2}\int_{\infty}^{\infty} \frac{(-x)^{s-1}}{e^x-1} dx$$
with explanation what path of integration he meant - depicted on the picture:
His motivation was to obtain a formula for $\zeta(s)$ that would be valid for all complex values of $s$.
1. But what was the reason behind choosing this particular integral $(2)$ that differs from $(1)$ only in replacing $x$ with $(-x)$ in one place?
2. And what was the reason to choose the particular integration path?
3. How is it evident that the new integral should be convergent in the whole complex plane?