To integrate the function $f(x) = 6x^4 - 9x^2 - \sqrt{x}$, we will integrate term by term.

Step 1: Rewrite the function in terms of powers of $x$

Recall that $\sqrt{x} = x^{1/2}$, so we rewrite the function as: $f(x) = 6x^4 - 9x^2 - x^{1/2}$

Step 2: Apply the power rule of integration

The power rule for integration is: $\int x^n \, dx = \frac{x^{n+1}}{n+1} + C \quad \text{(for } n \neq -1\text{)}$

Now, apply this rule to each term in the function.

1. Integrating $6x^4$:

[ \int 6x^4 , dx = 6 \cdot \frac{x^{4+1}}{4+1} = 6 \cdot \frac{x^5}{5} = \