A function $f$ is to be found out such that its derivative $f'(x)$ is given by $$f'(x)= \frac{192 x^3}{2+ \sin^4\pi x} \forall x \in \mathbb R$$
The value of the function at $x=0$ is $\frac{1}{2}$.
I have absolutely no idea on how to integrate $f'(x)$. Is it possible to find $f(x)$ at all? How can this integral be evaluated?
If you are looking for a closed form expression for the integral, then I would guess that there is no such closed form in elementary functions.
But if you are simply looking for an antiderivative $f$ of $f'$ such that $f(0) = 1/2$, then the unique such function is $$ f(x) = \frac{1}{2} + \int_0^x \frac{192 t^3}{2 + \sin^4 \pi t} dt.$$ For any value of $x$, it is possible to numerically estimate $f(x)$ up to any desired accuracy.