https://file.aiquickdraw.com/mathbot/file/1724071626892gbs63bp0.jpg

To solve the integral 

\[
\int (3x^4 + x^2 + 2) \, dx
\]

we can integrate each term separately.

### Step-by-Step Solution:

1. **Integrate the first term:**
   \[
   \int 3x^4 \, dx
   \]
   Use the power rule of integration: \(\int x^n \, dx = \frac{x^{n+1}}{n+1}\).

   \[
   \int 3x^4 \, dx = 3 \cdot \frac{x^{5}}{5} = \frac{3x^5}{5}
   \]

2. **Integrate the second term:**
   \[
   \int x^2 \, dx
   \]
   Again, use the power rule:
   \[
   \int x^2 \, dx = \frac{x^{3}}{3}
   \]

3. **Integrate the third term:**
   \[
   \int 2 \, dx
   \]
   Since \(2\) is a constant:
   \[
   \int 2 \, dx = 2x
   \]

### Combine the Results:

Putting everything together, the integral is:
\[
\int (3x^4 + x^2 + 2) \, dx = \frac{3x^5}{5} + \frac{x^3}{3} + 2x + C
\]
where \(C\) is the constant of integration.

---

Would you like any further details or have any questions about this solution?

Here are 5 related questions you might find interesting:
1. How do you apply the power rule to integrate polynomials with negative exponents?
2. What is the importance of the constant of integration \(C\) in indefinite integrals?
3. How would you integrate a function with a fractional exponent, like \(x^{\frac{3}{2}}\)?
4. What are some common mistakes to avoid when integrating polynomial functions?
5. How do you verify the result of an indefinite integral by differentiating it?

**Tip:** When integrating polynomials, always apply the power rule carefully and remember to include the constant of integration \(C\) at the end.

Learn how to integrate 3x^4 + x^2 + 2 with a detailed step-by-step solution. This problem covers integral calculus concepts including the power rule of integration and the constant of integration. Suitable for advanced high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation