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

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.

Integrate the function f(x) = 6x^4 - 9x^2 - square root of x

Learn how to integrate the function 6x^4 - 9x^2 - √x with a detailed step-by-step guide. This problem covers key calculus concepts such as polynomial integration, the power rule, and handling fractional exponents. Suitable for students in advanced high school calculus (Grades 11-12).

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