Expectation and Even PDF

By Definition of Expectation of Random Variable:

$$E(X)=\int_{-\infty}^{\infty}\,x\,f_X(x)\,dx$$

If $f_X(x)$ is Even, Then $E(X)=0$, Provided The Integral Exists, An Exception Being $\textbf{Cauchy Distribution}$ , Whose Mean Doesn't Exist.

Now the Question is, Will the PDF of Random Variable is Even ,When $E(X)=0$, I am Unable to Solve/Prove This. I Encourage all of you to Solve This...