General antiderivative of a power-function sum

Rewrite $f(x) = 9x^{-1/2} - 2x^{-3}$ term by term so the power rule can be applied. For the first term, increase the exponent by 1: $-1/2+1 = 1/2$, so $\int 9x^{-1/2}\,dx = 9\cdot\dfrac{x^{1/2}}{1/2} = 18x^{1/2}$. For the second term, $-3+1=-2$, so $\int -2x^{-3}\,dx = -2\cdot\dfrac{x^{-2}}{-2} = x^{-2}$.

Adding these results together and including the arbitrary constant of integration gives the general antiderivative $F(x) = 18x^{1/2} + x^{-2} + C.$ The $+C$ is essential since any constant has derivative zero, so infinitely many functions share this same derivative $f(x)$.