To integrate the function $f(x) = 6x^4 - 9x^2 - \sqrt{x}$, we will integrate each term separately. Let's break it down step by step:

Step 1: Break down the function

$f(x) = 6x^4 - 9x^2 - \sqrt{x}$

Step 2: Apply the power rule to each term

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

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

2. Integrating $-9x^2$: $\int -9x^2 \, dx = -9 \cdot \frac{x^{2+1}}{2+1} = -9 \cdot \frac{x^3}{3} = -3x^3$

3. Integrating $-\sqrt{x}$ or $-x^{1/2}$: $\int -x^{1/2} \, dx = -\frac{x^{\frac{1}{2} + 1}}{\frac{1}{2} + 1} = -\frac{x^{3/2}}{\frac{3}{2}} = -\frac{2}{3}x^{3/2}$

Step 3: Combine the results

Now, combining all the integrated terms, we get: $\int f(x) \, dx = \frac{6x^5}{5} - 3x^3 - \frac{2}{3}x^{3/2} + C$

where $C$ is the constant of integration.

Final answer:

$\int (6x^4 - 9x^2 - \sqrt{x}) \, dx = \frac{6x^5}{5} - 3x^3 - \frac{2}{3}x^{3/2} + C$

Would you like further details on any of the steps?

Related Questions:

1. How do you integrate functions involving roots of higher powers?
2. What is the power rule for integration, and when does it fail?
3. How would you integrate a function like $x^{-3/2}$?
4. What are the differences between definite and indefinite integrals?
5. How do you check if the integration result is correct?

Tip:

Always remember to add the constant of integration, $C$, when performing indefinite integrals.