1. The problem is to find the integral in simplest form of the function $$\int \left( \frac{8x^3}{3} - \frac{1}{2\sqrt{x}} - 5 \right) dx$$. 2. Recall the integral rules: - The integral of $x^n$ is $$\frac{x^{n+1}}{n+1} + C$$ for $n \neq -1$. - The integral of a constant $a$ is $$ax + C$$. 3. Rewrite the terms for easier integration: - $$\frac{8x^3}{3}$$ stays as is. - $$\frac{1}{2\sqrt{x}} = \frac{1}{2} x^{-\frac{1}{2}}$$. - The constant term is $$-5$$. 4. Integrate each term separately: - $$\int \frac{8x^3}{3} dx = \frac{8}{3} \int x^3 dx = \frac{8}{3} \cdot \frac{x^{4}}{4} = \frac{8}{3} \cdot \frac{x^{4}}{4} = \frac{2x^{4}}{3}$$. - $$\int -\frac{1}{2} x^{-\frac{1}{2}} dx = -\frac{1}{2} \int x^{-\frac{1}{2}} dx = -\frac{1}{2} \cdot \frac{x^{\frac{1}{2}}}{\frac{1}{2}} = -\frac{1}{2} \cdot 2x^{\frac{1}{2}} = -x^{\frac{1}{2}} = -\sqrt{x}$$. - $$\int -5 dx = -5x$$. 5. Combine all integrated parts and add the constant of integration $C$: $$\frac{2x^{4}}{3} - \sqrt{x} - 5x + C$$. 6. Therefore, the integral in simplest form is: $$\int \left( \frac{8x^3}{3} - \frac{1}{2\sqrt{x}} - 5 \right) dx = \frac{2x^{4}}{3} - \sqrt{x} - 5x + C$$.