trend of an integral function I have an array $f(x)$, which exhibits (at least in the first frequencies) a $x^{-5/3}$ dependence. This array theoretically has this relation with the array $g(x)$:
$f(x)=\frac{d g(x)}{dx}$
I wanted to find $g(x)$ so I did the cumulative trapezoidal numerical integration of $f(x)$. As you can see in the figure attached I have found this trend for $g(x)$.
[![enter image description here][1]][1]
I was expecting a trend of $x^{-2/3}$, since following the integral rule, the power should be $x^{(-5/3)+1}$
Can you explain me why this rule is not applied?
$$  \int x^{-5/3} \,\mathrm{d}x = \frac{x^{-2/3}}{-2/3} + C = \frac{-3}{2}x^{-2/3} + C  \text{.}  $$

Notice the minus sign.  This is a function that increases, asymptotically approaching the constant of integration.  Taking $C = 10\,000$ to roughly match your data and zooming in a bit,
$$  \int x^{-5/3} \,\mathrm{d}x = \frac{x^{-2/3}}{-2/3} + C = \frac{-3}{2}x^{-2/3} + C  \text{.}  $$

Notice the minus sign.  This is a function that increases, asymptotically approaching the constant of integration.  Taking $C = 10\,000$ to roughly match your data and zooming in a bit,