We are asked to find the indefinite integral of the function $5x^4 - 3x^2 + \frac{3\sqrt{x}}{2}$ with respect to $x$.

1. Problem Description

We are asked to find the indefinite integral of the function $5x^4 - 3x^2 + \frac{3\sqrt{x}}{2}$ with respect to $x$.

2. Solution Steps

We will integrate term by term. Recall that the power rule for integration is:

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

First, we integrate $5x^4$:

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

Second, we integrate $-3x^2$:

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

Third, we integrate $\frac{3\sqrt{x}}{2}$. Note that $\sqrt{x} = x^{\frac{1}{2}}$.

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

Therefore, the integral is:

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

3. Final Answer

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