On Wikipedia, the article about the power rule states that if $f:\mathbb{R}\to\mathbb{R}$ is given by $f(x)=x^n$, then
$$ f'(x)=nx^{n-1} $$
provided that $f$ is differentiable at $x$, $n$ is a constant, and $x$ is a variable. The article then goes on to say that applying the fundamental theorem of calculus allows us to derive the power rule for integration:
$$ \int x^ndx=\frac{x^{n+1}}{n+1}+C $$
I don't understand how the fundamental theorem of calculus is relevant here. I have often read that the fundamental theorem of calculus tells us that 'integration is the reverse process to differentiation', but I assumed that this was referring to definite integration. For example, the first fundamental theorem of calculus tells us that
$$ \frac{d}{dx}\int_{a}^{x}f(t)dt=f(x) $$
This is a crucially important result that relates finding the area with finding the gradient, but (as far as I understand) says nothing about indefinite integrals. Surely the fact that
$$ \int\left(\frac{d}{dx}x^n\right)dx=x^n+C $$
is pretty much true by definition, and is by no means an interesting result that deserves the name 'fundamental'. Indefinite integration (or finding the antiderivative) means 'what differentiates to the integrand'? The fundamental theorem of calculus is only interesting because it shows the link between finding the area and finding the antiderivative, which is not immediately apparent. However, because the power rule only deals with indefinite integration, can't it be proved with no appeal to the fundamental theorem of calculus? Here is an attempt:
$$ \frac{d}{dx}\left(\frac{x^{n+1}}{n+1}+C\right)=\frac{1}{n+1}\frac{d}{dx}(x^{n+1})=\frac{n+1}{n+1}x^n=x^n $$
The indefinite integral of $x^n$ means 'what function (or family of functions) differentiate to $x^n$?' Therefore,
$$ \int x^ndx=\frac{x^{n+1}}{n+1}+C $$