\(a_0=\dfrac{1}{\pi}\int_{-\pi}^{0}(3\pi+2x)\,dx+\dfrac{1}{\pi}\int_0^\pi(\pi+2x)\,dx\). First: \(\dfrac{1}{\pi}[3\pi x+x^2]_{-\pi}^0=\dfrac{1}{\pi}[0-(-3\pi^2+\pi^2)]=\dfrac{2\pi^2}{\pi}=2\pi\). Second: \(\dfrac{1}{\pi}[\pi x+x^2]_0^\pi=\dfrac{1}{\pi}(\pi^2+\pi^2)=2\pi\). Total: \(a_0=4\pi\).