The problem asks to find the indefinite integral of the function $5x^4 - 3x^2 + \frac{3\sqrt{x}}{2}$.

1. Problem Description

The problem asks to find the indefinite integral of the function $5x^4 - 3x^2 + \frac{3\sqrt{x}}{2}$.

2. Solution Steps

We need to evaluate the integral:

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

We can break the integral into three separate integrals:

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

We can pull the constants out of the integrals:

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

Recall the power rule for integration:

$\int x^n dx = \frac{x^{n+1}}{n+1} + C$

Applying the power rule to each term:

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

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

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

Putting it all together:

$x^5 - x^3 + x^{\frac{3}{2}} + C$

3. Final Answer

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