Is there a continuous, smooth function $g(x)$, where
- On $ x \in[0,0.5], \, c\leq g(x)<0.5$, with $c$ as a constant $<0.5$, but on $x \in [0.5, \infty], \, \lim_ {x \to \infty} g(x)=0.5$ (hopefully with some kind of parameter $k$ I could adjust to "speed up" the convergence).
- Also, $g(F(x)^2)$, where $F(x)$ is a Fourier series of the form $c+\sum_{i=1}^n a_i \sin(b_i x)$, must have a closed-form integral (no special functions like $\mathrm{Si}(x) $ or $\mathrm{Li}(x)$, but functions like $|x|$ are allowed).
For example, an acceptable answer I'm asking could be a smooth version of $\min(x,0.5)$, but with the second requirement; but it's not necessary.
EDIT: I found $g(x)=\frac12 \left(-\frac1{50x+1}+1 \right)$, though using Mathematica to integrate $g(F(x)^2)$ with $F(x)=6+\sin(3x)+\sin(5x)+\sin(7x)$ as a test was very messy (I'm not even sure it's correct). Is there a better way of integrating this function, or another $g(x)$ that has a cleaner integral?