https://file.aiquickdraw.com/mathbot/file/1725999590234da6odrcp.jpg

The given integral is:

\[
\int \frac{x^2}{x^2 + 6x + 12} \, dx
\]

Let's solve this step by step.

### Step 1: Simplify the denominator
The denominator \( x^2 + 6x + 12 \) can be rewritten by completing the square.

\[
x^2 + 6x + 12 = (x + 3)^2 + 3^2
\]

Thus, the integral becomes:

\[
\int \frac{x^2}{(x + 3)^2 + 9} \, dx
\]

### Step 2: Divide the numerator
We can split the numerator \( x^2 \) as:

\[
x^2 = (x + 3)^2 - 6x - 9
\]

Substituting this into the integral:

\[
\int \frac{(x + 3)^2 - 6x - 9}{(x + 3)^2 + 9} \, dx
\]

This can be split into two separate integrals:

\[
\int \frac{(x + 3)^2}{(x + 3)^2 + 9} \, dx - \int \frac{6x + 9}{(x + 3)^2 + 9} \, dx
\]

### Step 3: Simplify and solve each integral

1. For the first part:

\[
\int \frac{(x + 3)^2}{(x + 3)^2 + 9} \, dx
\]

This simplifies to:

\[
\int 1 \, dx = x
\]

2. For the second part:

\[
\int \frac{6(x + 3)}{(x + 3)^2 + 9} \, dx
\]

This can be solved using a substitution \( u = x + 3 \), and it becomes a standard arctangent form.

The result of this part is:

\[
2 \ln |(x + 3)^2 + 9| + C
\]

### Final Solution:

The integral simplifies to:

\[
x - 2 \ln |(x + 3)^2 + 9| + C
\]

This is the solution to the given integral.

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

Here are 5 related questions:
1. How do you complete the square for quadratic expressions?
2. What are the standard forms of integrals involving arctangent?
3. How do you handle integrals that involve splitting the numerator and denominator?
4. Can partial fraction decomposition be used for integrals like this one?
5. What other techniques can simplify integrals with quadratic terms?

**Tip:** When working with integrals involving quadratics, completing the square helps simplify complex expressions!

Solve the integral: ∫(x²/(x² + 6x + 12)) dx

This article provides a detailed solution for the integral ∫(x²/(x² + 6x + 12)) dx, covering key techniques such as completing the square and substitution for arctangent forms. Suitable for students in advanced calculus or integration courses.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation