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?

The power rule for integration says that
$$
\int x^n dx = \frac{x^{n+1}}{n+1}
$$
Why doesn't that work for $\int [f(x)]^n dx$? Because $f(x)$ does not match $dx$. The proper use of the power rule would be
$$
\int f(x)^n df(x) = \frac{f(x)^{n+1}}{n+1}
$$
But we can still get something useful out of this by using, as you put it, some sort of chain rule. Namely, the chain rule, which tells us $df(x) = f'(x) dx$, which makes the above into
$$
\int f(x)^n f'(x)dx = \frac{f(x)^{n+1}}{n+1}
$$
As you may have noticed, this is just a  $u$-substitution with $u = f(x)$. In fact, $u$-substitution is in general equivalent to the chain rule, and thus is the sort of chain rule you describe in the question.
