Evaluate the definite integral $\int_0^{2016} x (x-1)(x-2)(x-3)... (x-2016)\,dx$ 
$$\int_0^{2016} x (x-1)(x-2)(x-3)... (x-2016)\,dx$$ 

I have tried integration by parts using calling $f (x)=(x-1)(x-2)... (x-2016)$ but it doesn't help. I can't think of anything else. Please help.

Note that 
$$\scriptsize\begin{align}
\int_0^2 x(x-1)(x-2)dx&=\int_{-1}^1 (x-1)x(x+1)dx&=0\\
\int_0^4 x(x-1)(x-2)(x-3)(x-4)dx&=\int_{-2}^2 (x-2)(x-1)x(x+1)(x+2)dx&=0\\
&\vdots\\
\int_0^{2016} x(x-1)(x-2)(x-3)\cdots (x-2016)dx&=\int_{-1008}^{1008} (x-1008)(x-1007)\cdots x\cdots (x+1007)(x+1008)dx&=\color{red}0\\
\end{align}$$
The result follows from the antisymmetric nature of the curve
$\scriptsize(x-a)(x-a+1)\cdots x\cdots (x+a-1)(x+a)$.
