\int_0^{2\pi}f(x)\,dx=\int_0^\pi f(x)\,dx+\int_\pi^{2\pi}f(x)\,dx=\int_0^\pi f(x)\,dx+\int_{-\pi}^0f(x+2\pi)\,dx=\int_0^\pi f(x)\,dx+\int_{-\pi}^0 f(x)\,dx=\int_{-\pi}^{\pi} f(x)\,dx