Recall that a function $f$ is called odd if $f(x)=-f(-x)$ for all $x,$ and a function $g$ is called even if $f(x)=f(-x).$

$($a$)$ Show that, if $f(x)$ is continuous and odd, then ${\displaystyle \underset{-a}{\overset{a}{}}}f(x)dx=0.$

$($b$)$ Show that, if $g(x)$ is continuous and even, then ${\displaystyle \underset{-a}{\overset{a}{}}}g(x)dx=2{\displaystyle \underset{0}{\overset{a}{}}}g(x)dx.$

$($c$)$ Show that any function can be written as the sum of an odd function and an even function. Hint: Given a function $f,$ what can we say about $h(x)=\frac{f(x)+f(-x)}{2}?$

$($d$)$ Let kinR be a fixed constant. Show that, if $f$ is an odd continuous function, and $g$ is an even continuous function, then

${\displaystyle \underset{-a}{\overset{a}{}}}\frac{g(x)}{1+k^{f(x)}}dx={\displaystyle \underset{0}{\overset{a}{}}}g(x)dx$

$($e$)$ Evaluate ${\displaystyle \underset{-}{\overset{}{}}}\frac{x^{2024}}{1+{}^{x^{3}cosx}}dx.$