General antiderivative of a power-function sum

Rewrite $h(x)$ using exponent notation: $h(x) = 3x^{-2} + 4x^{2/3}$. This makes it clear both terms are simple powers of $x$, so the power rule for antiderivatives can be applied to each.

For the first term, $\int 3x^{-2}\,dx = 3\cdot\frac{x^{-1}}{-1} = -3x^{-1} = -\frac{3}{x}$. For the second term, $\int 4x^{2/3}\,dx = 4\cdot\frac{x^{5/3}}{5/3} = \frac{12}{5}x^{5/3}$. Adding these and including the arbitrary constant gives $H(x) = -\frac{3}{x} + \frac{12}{5}x^{5/3} + C$.