I calculated the integral of this complex function $(-1)^x$:
$$\int_{0}^{1}(-1)^x dx = \frac{(-1)^x}{\mathrm{Ln}(-1)}\bigg|_{0}^{1}=\frac{(-1)^x}{\ln|-1|+i\theta(-1)}\bigg|_{0}^{1}=\frac{(-1)^x}{i\pi}\bigg|_{0}^{1}=-\frac{2}{i\pi}=\frac{2i}{\pi}$$ This is the plot of $(-1)^x$:
Does this value have any geometric meaning, like could be the area of something related to the graph multiplied by $i$?
(Also, please verify my calculation. I used the principal branch of logarithm, but I am not sure if that is what I am supposed to do in this case).
Thanks!
One definition of $(-1)^x$ is $e^{i \pi x}$; this is the plot shown in the OP. If you keep this definition over the whole domain $[0,1]$, then
$$\int_0^1 (-1)^x dx = \int_0^1 \cos(\pi x) dx + i \int_0^1 \sin(\pi x) dx = \frac{2i}{\pi}.$$
For completeness, in general $(-1)^x$ is the multivalued function $\exp(x \ln(-1))=\{ \exp(x(2k+1) i \pi) : k \in \mathbb{Z} \}$