1. The problem is to find the original function $F(x)$ given its derivative $F'(x) = 3 + \frac{5x^2 + 2}{x^{1/2}}$. 2. Rewrite the derivative to simplify the integral: $$F'(x) = 3 + 5x^{2} \cdot x^{-1/2} + 2x^{-1/2} = 3 + 5x^{3/2} + 2x^{-1/2}$$ 3. Integrate each term separately: $$F(x) = \int 3 \, dx + \int 5x^{3/2} \, dx + \int 2x^{-1/2} \, dx$$ 4. Compute each integral: - $\int 3 \, dx = 3x$ - $\int 5x^{3/2} \, dx = 5 \cdot \frac{x^{5/2}}{5/2} = 5 \cdot \frac{2}{5} x^{5/2} = 2x^{5/2}$ - $\int 2x^{-1/2} \, dx = 2 \cdot \frac{x^{1/2}}{1/2} = 2 \cdot 2 x^{1/2} = 4x^{1/2}$ 5. Combine the results and add the constant of integration $C$: $$F(x) = 3x + 2x^{5/2} + 4x^{1/2} + C$$ This is the original function $F(x)$ whose derivative is given.