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

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.

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

Learn how to integrate the function f(x) = 6x^4 - 9x^2 - √x step-by-step. This solution demonstrates the application of the power rule for integration, covering polynomials and radicals. Suitable for Grades 11-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation