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

Step 1: Break down the function

The function is composed of three terms: $f(x) = 6x^4 - 9x^2 - \sqrt{x}$ We can write the square root of $x$ as $x^{1/2}$.

So the function becomes: $f(x) = 6x^4 - 9x^2 - x^{1/2}$

Step 2: Apply the power rule for integration

The power rule for integration states: $\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$ where $n \neq -1$.

Step 3: Integrate each term

1. For $6x^4$: $\int 6x^4 \, dx = 6 \cdot \frac{x^{5}}{5} = \frac{6x^5}{5}$

2. For $-9x^2$: $\int -9x^2 \, dx = -9 \cdot \frac{x^{3}}{3} = -3x^3$

3. For $-x^{1/2}$: $\int -x^{1/2} \, dx = -\frac{2x^{3/2}}{3}$

Step 4: Combine the results

The integral of $f(x)$ is: $\int f(x) \, dx = \frac{6x^5}{5} - 3x^3 - \frac{2x^{3/2}}{3} + C$ where $C$ is the constant of integration.

Final Answer:

$\boxed{\int f(x) \, dx = \frac{6x^5}{5} - 3x^3 - \frac{2x^{3/2}}{3} + C}$

Would you like more details or have any questions?

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Here are 5 related questions for further exploration:

1. How would you integrate the function $f(x) = 4x^3 - 7x + \sqrt{x}$?
2. What is the integral of a general polynomial function $ax^n$?
3. How do you find the integral of a function involving $\sqrt{x}$ or $x^{1/n}$?
4. Can you apply integration to a function with negative powers of $x$?
5. How do you compute definite integrals, such as $\int_1^2 (6x^4 - 9x^2 - \sqrt{x}) \, dx$?

Tip: When integrating functions with roots, express the root as a fractional exponent for easier application of the power rule.