General antiderivative of a sum of power functions

Rewrite $h(x) = 3x^{1/2} - 2x^{-1/2}$. Apply the reverse power rule term by term: $\int 3x^{1/2}\,dx = 3 \cdot \dfrac{x^{3/2}}{3/2} = 2x^{3/2}$, and $\int -2x^{-1/2}\,dx = -2 \cdot \dfrac{x^{1/2}}{1/2} = -4x^{1/2}$.

Adding these results and including a single arbitrary constant gives the general antiderivative $H(x) = 2x^{3/2} - 4x^{1/2} + C$. You can check this by differentiating: $H'(x) = 2\cdot\frac{3}{2}x^{1/2} - 4\cdot\frac{1}{2}x^{-1/2} = 3x^{1/2} - 2x^{-1/2} = h(x)$, confirming the answer.

The answer is C. In the solution to part a, the general antiderivative was written as H(x) = 2x^{3/2} - 4x^{1/2} + C, where C denotes the arbitrary constant of integration. This constant accounts for the fact that any two antiderivatives of the same function differ only by a constant, since the derivative of a constant is zero.