Chọn phương án D.

Phương trình hoành độ giao điểm: $$\begin{aligned}&\,x^2-x+5=-2x^3+x^2+x+5\\ \Leftrightarrow&\,2x^3-2x=0\\ \Leftrightarrow&\,\left[\begin{array}{l}
x=0\\ x=\pm1.
\end{array}\right.\end{aligned}$$

$\begin{aligned}\Rightarrow S&=\displaystyle\int\limits_{-1}^{0}\left(2x^3-2x\right)\mathrm{\,d}x+\displaystyle\int\limits_{0}^{1}\left(2x-2x^3\right)\mathrm{\,d}x\\ &=\left(\dfrac{x^4}{2}-x^2\right)\bigg|_{-1}^0+\left(x^2-\dfrac{x^4}{2}\right)\bigg|_0^1\\ &=\dfrac{1}{2}+\dfrac{1}{2}=1.\end{aligned}$