What is the value of $\int_0^1\pi e^{i\pi t}dt$?

Question

What is the integration of the following equation?

$$\int_0^1Z(t)dt =\int_0^1\pi e^{i\pi t}dt$$

Answer 1

The integral of $\int_0^1 \pi e^{i\pi t}\ dt$ Is by u substitution with $u = i\pi t$ and $du/dt= i\pi$ is equal to (also substituting the bounds) $\frac{\pi}{i\pi}\int_0^{i\pi}e^udu$. Since the antiderivative of $e^u$ is $e^u$, again substituting in the bounds and using Euler's identity, we get $-1/i-1/i=2i$.