The problem is to find the indefinite integral of the expression $(5x^4 - 3x^2 + \frac{3\sqrt{x}}{2})$ with respect to $x$.

1. Problem Description

The problem is to find the indefinite integral of the expression $(5x^4 - 3x^2 + \frac{3\sqrt{x}}{2})$ with respect to $x$.

2. Solution Steps

We need to evaluate $\int (5x^4 - 3x^2 + \frac{3\sqrt{x}}{2}) dx$.

We can split the integral into three parts:

$\int 5x^4 dx - \int 3x^2 dx + \int \frac{3\sqrt{x}}{2} dx$.

We know that $\int x^n dx = \frac{x^{n+1}}{n+1} + C$, where $C$ is the constant of integration.

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

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

Third, $\int \frac{3\sqrt{x}}{2} dx = \frac{3}{2} \int \sqrt{x} dx = \frac{3}{2} \int x^{\frac{1}{2}} dx = \frac{3}{2} \cdot \frac{x^{\frac{1}{2}+1}}{\frac{1}{2}+1} + C_3 = \frac{3}{2} \cdot \frac{x^{\frac{3}{2}}}{\frac{3}{2}} + C_3 = \frac{3}{2} \cdot \frac{2}{3} x^{\frac{3}{2}} + C_3 = x^{\frac{3}{2}} + C_3$.

Putting it all together:

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

3. Final Answer

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