The problem asks us to evaluate the definite integral $\int (5x^4 - 3x^2 + \frac{3\sqrt{x}}{2}) dx$.

1. Problem Description

The problem asks us to evaluate the definite integral $\int (5x^4 - 3x^2 + \frac{3\sqrt{x}}{2}) dx$.

2. Solution Steps

We can integrate each term separately using the power rule for integration. The power rule states that:

$\int x^n dx = \frac{x^{n+1}}{n+1} + C$, where $n \ne -1$.

First, consider the term $5x^4$. Using the power rule, we get

$\int 5x^4 dx = 5 \int x^4 dx = 5 \cdot \frac{x^{4+1}}{4+1} + C_1 = 5 \cdot \frac{x^5}{5} + C_1 = x^5 + C_1$.

Next, consider the term $-3x^2$. Using the power rule, we get

$\int -3x^2 dx = -3 \int x^2 dx = -3 \cdot \frac{x^{2+1}}{2+1} + C_2 = -3 \cdot \frac{x^3}{3} + C_2 = -x^3 + C_2$.

Finally, consider the term $\frac{3\sqrt{x}}{2}$. We can rewrite $\sqrt{x}$ as $x^{1/2}$. Then we have

$\int \frac{3\sqrt{x}}{2} dx = \frac{3}{2} \int x^{1/2} dx = \frac{3}{2} \cdot \frac{x^{(1/2)+1}}{(1/2)+1} + C_3 = \frac{3}{2} \cdot \frac{x^{3/2}}{3/2} + C_3 = \frac{3}{2} \cdot \frac{2}{3} x^{3/2} + C_3 = x^{3/2} + C_3$.

Adding all the integrated terms together, we have

$\int (5x^4 - 3x^2 + \frac{3\sqrt{x}}{2}) dx = x^5 - x^3 + x^{3/2} + C$, where $C = C_1 + C_2 + C_3$.

3. Final Answer

$x^5 - x^3 + x^{3/2} + C$