Let u=eˣ, so dx=du/u and 1/(eˣ+e⁻ˣ)dx=du/(1+u²). Integration over the real line becomes ∫₀∞du/(1+u²)=π/2. Thus the total probability is (q/π)(π/2)=q/2. Setting it to one gives q=2, A, also ensuring nonnegative density.
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