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please dont explain much only show all working out n all steps

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

1. **Integral of \( 6x^4 \):**
   \[
   \int 6x^4 \, dx = 6 \int x^4 \, dx = 6 \left( \frac{x^5}{5} \right) = \frac{6x^5}{5}
   \]

2. **Integral of \( -9x^2 \):**
   \[
   \int -9x^2 \, dx = -9 \int x^2 \, dx = -9 \left( \frac{x^3}{3} \right) = -3x^3
   \]

3. **Integral of \( -\sqrt{x} \):**
   \[
   \int -\sqrt{x} \, dx = - \int x^{1/2} \, dx = - \left( \frac{x^{3/2}}{\frac{3}{2}} \right) = - \frac{2}{3} x^{3/2}
   \]

Combining all these results, the indefinite integral of \( f(x) \) is:
\[
\int f(x) \, dx = \int (6x^4 - 9x^2 - \sqrt{x}) \, dx = \frac{6x^5}{5} - 3x^3 - \frac{2}{3} x^{3/2} + C
\]

Where \( C \) is the constant of integration.

### Final Answer:
\[
\boxed{\frac{6x^5}{5} - 3x^3 - \frac{2}{3} x^{3/2} + C}
\]

Would you like any more details or have any questions?

Here are some related questions you might find useful:
1. How do you find the definite integral of a polynomial function over a specific interval?
2. What are the rules for integrating functions involving fractional exponents?
3. How does the power rule for integration work?
4. What is the importance of the constant of integration?
5. Can you show the integration steps for a trigonometric function like \( \sin(x) \)?

**Tip:** When integrating, always ensure you adjust the exponent correctly and divide by the new exponent.

Integrate the function f(x) = 6x^4 - 9x^2 - √x

Learn how to integrate the function 6x^4 - 9x^2 - √x step by step using the power rule for integration. This problem covers key calculus concepts, including integrating polynomial functions and functions with fractional exponents. Suitable for students in Grades 11-12 or early university calculus.

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