Calculus

$a_0 = \frac{1}{\pi} \int_{-\pi}^\pi x^2 dx = \frac{2}{\pi} \left[\frac{x^3}{3}\right]_0^\pi = \frac{2\pi^2}{3}$. Therefore $\frac{a_0}{2} = \frac{\pi^2}{3}$, so $k = 3$.