calculus - Is there a $g$ such that $intlimits_{-a}^a frac{f(x)}{g(x)} mathrm dx = intlimits_0^a f(x) mathrm dx$ for *odd* function $f$?

It was shown that if $f$ is continuous and even on $[−a,a], a>0$, then $\int\limits_{-a}^a \frac{f(x)}{1+e^{x}} \mathrm dx = \int\limits_0^a f(x) \mathrm dx$.

I wonder are there any similar property holds if $f$ is odd? So my question is as follow:

If $f$ is continuous and odd on $[−a,a], a>0$, is there a function $g$ such that $\int\limits_{-a}^a \frac{f(x)}{g(x)} \mathrm dx = \int\limits_0^a f(x) \mathrm dx$?