Why aren't all functions considered a power function of 1 and integrated using some sort of chain rule? When integrating, say, $f(x) = x^{3} - 2x$, why do we go straight to integrating each term independently? Why is it not considered an implicit function of the power one and integrated as so?
I'm not sure if that's a valid feat since I'm a beginner, but I was thinking: $\int \left [ f(x) \right ] ^{^{1}} dx$ would be $\frac{f(x)^{2}}{2}$ as another example to the usual: $\int f(x)^{n} dx  = \frac{f(x)^{n+1}}{n+1}$ where $n \neq -1 $.
As for what's inside, the $f(x)$, as I said I'm a beginner, but I know of u-substitution and maybe we could use it and work it out from there?
Is that possible or just plain wrong?

You forget the chain rule.  Indeed,
$$\frac d{dx}\frac{y^2}2=y\frac{dy}{dx}\ne y$$
Which is why integration is so much harder than differentiation.
However, it is true that
$$\int[f(x)]^nf'(x)\ dx=\frac{[f(x)]^{n+1}}{n+1}+c$$
Which should follow from u-substitution or differentiating both sides.
