I have the density function $$f(x)= \frac {e^x}{(e^x+1)^2}.$$ The integral from $-\infty$ to $\infty$ is $1$ so it is indeed a density function. The expected value of the function is $0$ and the variance is $\pi^2/3$.
My goal is to set the variance to $1$ but leaving the expected value at $0$ and of course the Integral of $f(x)$ from $-\infty$ to $\infty$ should stay at $1$ in order to have a density. But I am clueless of how to do it. Also is there a general way to perform this kind of task for any given density function?
Hint: If $g(x)$ is a density function, then so is $$ a\cdot g(ax) $$ for any $a\in (0, \infty)$.