= Solution
A <probability density function> must be nonnegative. The printed condition $s\ne0$ is therefore too weak: for $s<0$ the given prefactor makes the function negative everywhere. The <location-scale form of the logistic distribution> requires \b[$s>0$], which is assumed in the remaining parts.
Set $t=(x-\mu)/(2s)$, so $dx=2s\,dt$. The <normalization of a probability distribution> follows from the <antiderivative> of the squared <hyperbolic secant>:
$$
\int_{-\infty}^{\infty}f(x)\,dx
=\frac12\int_{-\infty}^{\infty}\operatorname{sech}^2t\,dt
=\frac12\big[\tanh t\big]_{-\infty}^{\infty}
=\boxed{1}.
$$
For completeness, the same expression with $s<0$ integrates to $-1$, since the transformed limits reverse. If a signed scale were desired, replacing its prefactor by $1/(4|s|)$ would give a valid density, but that is a correction to the printed formula.
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