Why does the Dirichlet Kernel $D_N$, the sum of exponentials gives $\frac{1}{2\pi}\int_{-\pi}^{\pi}D_N(x)~dx =1$?

Question

Let the Dirichlet kernel be defined by

$$D_N(x) = \sum_{n=-N}^{N}e^{inx}$$

I found a proposition that says that for an integer $j$,

$$\int_{0}^{1}e^{i2\pi jx}dx = \begin{cases} 1, ~~~j=0\\0,~~~j\neq 0 \end{cases} $$

and that this can easily be verified with calculus. However when I computed the integral, I got $$\int_{0}^{1}e^{i2\pi jx}dx = -\frac{i}{2\pi j}\bigg(e^{2\pi ij} - 1\bigg) $$

Why then isn't the integral undefined for $j=0$?