Integral of Even and Odd functions Can I get help to prove or Disprove that - 
$\int_{-a} ^a F_1(x)F_2(x)dx = 0 \ for \ a > 0$
when I know that $F_2$  is even function and $F_1$  is odd.
I would like to get the idea how to delete and extract Minus in integrals.
thanks!

I will replace your two functions with $f$ and $g$ for convenience. 
We have $h(x) = f(x)g(x) = f(-x) \cdot -g(-x) = -h(-x)$
Hence $$\int_{-a} ^a h(x) \ dx = \int_{-a} ^0 h(x) \ dx -  \int_{0} ^a h(-x) \ dx = \int_{-a} ^0 h(x) \ dx - \int_{-a} ^{0} h(x) \ dx = 0$$
