A number X is randomly selected from the interval $(-\pi,\ \pi)$, so that X is uniformly distributed over that interval. Which of these is the cumulative distribution function of X?

The interval has length $2\pi$, so the density is $f(x)=\dfrac{1}{2\pi}$ on it. Integrating from the lower end, $F(x)=\displaystyle\int_{-\pi}^{x}\frac{1}{2\pi}\,dt=\frac{x+\pi}{2\pi}$ for $-\pi<x<\pi$, which rises from 0 at $x=-\pi$ to 1 at $x=\pi$. The middle piece $\frac{1}{2\pi}$ is the density itself, not the distribution function; $\frac{x}{2\pi}$ gives $-\frac12$ rather than 0 at $x=-\pi$; and $\frac{x+2\pi}{2\pi}$ gives $\frac12$ at $x=-\pi$ and $\frac32$ at $x=\pi$, so neither of those runs from 0 to 1.