A calculus problem by ابراهيم فقرا

All the functions $x,1,\cos x, \sin x$ are orthogonal, meaning that $\int_{-\pi}^{\pi} f(x)g(x)dx=0$ , except for $x$ and $\sin x$ . For a real $c$ , we find that $\int_{-\pi}^{\pi} (x-c\sin x)^2\ dx=\frac{2\pi^3}{3}+c(c-4)\pi$ is minimal when $c=2$ , and the minimal value is $\frac{2}{3}\pi^3-4\pi \approx \boxed{8.1045}$ when we let $a=b=\Im c=0$ .