If $f(x)$ and $g(x)$ are two continuous functions defined on $[-a, a]$, then the value of $\int\limits_{-a}^a\{f(x)+f(-x)\}\{g(x)-g(-x)\} d x$, is

none of these

Clearly, $f(x)+f(-x)$ is an even function and $g(x)+g(-x)$ is an odd function.

∴  $\{f(x)+f(-x)\}\{g(x)-g(-x)\}$ is an odd function.

$\Rightarrow \int\limits_{-a}^a\{f(x)+f(-x)\}\{g(x)-g(-x)\} d x=0$