Integrate the function f(x) = 6x4 - 9x2 - √𝑥

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

1. **Integrate \( 6x^4 \):**
   \[
   \int 6x^4 \, dx = 6 \cdot \frac{x^5}{5} = \frac{6x^5}{5}
   \]

2. **Integrate \( -9x^2 \):**
   \[
   \int -9x^2 \, dx = -9 \cdot \frac{x^3}{3} = -3x^3
   \]

3. **Integrate \( -\sqrt{x} \) (which is \( -x^{1/2} \)):**
   \[
   \int -x^{1/2} \, dx = -\frac{2x^{3/2}}{3}
   \]

Now, combining all the results:
\[
\int f(x) \, dx = \frac{6x^5}{5} - 3x^3 - \frac{2x^{3/2}}{3} + C
\]
where \( C \) is the constant of integration.

### Final Answer:
\[
\int (6x^4 - 9x^2 - \sqrt{x}) \, dx = \frac{6x^5}{5} - 3x^3 - \frac{2x^{3/2}}{3} + C
\]

Would you like further details or have any questions?

Here are some related questions you might explore:
1. How do you apply integration to trigonometric functions?
2. What happens if the square root term had a higher exponent?
3. How does integration apply to complex functions?
4. What is the geometric interpretation of an integral?
5. How can you find definite integrals with specific limits?

**Tip**: Always check the powers of \( x \) before integrating, as using the power rule requires adjusting the exponent correctly.

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

This article provides a step-by-step solution to the integration of the function 6x^4 - 9x^2 - √x. Explore the power rule, the handling of square roots, and how each term is integrated. Suitable for students in Grades 11-12.

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