Find the integral

It is easy to see that

$\frac{\mathrm{d^{700}} }{\mathrm{d} x^{700}} \left [ (x^{2}-1)^{700} \right ] = 2^{700}700!P_{700}(x)$

and

$\sum_{k=0}^{4} (-1)^{k} \begin{pmatrix} 8\\ k \end{pmatrix} \begin{pmatrix} 16-2k\\ 8 \end{pmatrix} x^{8-2k} = 2^{8}P_{8}(x)$

where $P_{l}(x)$ stands for the $l$ -th Legendre Polynomial. As we know

$\int_{-1}^{1} P_{m}(x)P_{n}(x) \; \mathrm{d}x = \frac{2}{2m+1} \delta_{m,n}$

is the normalization integral, where $\delta_{m,n}$ stands for the Kronecker delta. Now it is easy to notice that if we denote the original integral by $O$ we get

$O = 2^{708} \times 700! \times \int_{-1}^{1} P_{700}(x)P_{8}(x) \; \mathrm{d}x = 2^{708} \times 700! \times 0 = 0$