https://file.aiquickdraw.com/mathbot/file/1726759837352q5luslkt.jpg

We are tasked with finding the indefinite integral using the substitution \( x = \tan(\theta) \). The integral is:

\[
\int \frac{x^2}{(1+x^2)^2} \, dx
\]

### Step-by-Step Solution

1. **Substitute \( x = \tan(\theta) \):**
   Using the given substitution \( x = \tan(\theta) \), we know the following:
   \[
   dx = \sec^2(\theta) \, d\theta
   \]
   and
   \[
   1 + x^2 = 1 + \tan^2(\theta) = \sec^2(\theta)
   \]

2. **Rewrite the integral:**
   Substituting into the integral:

   \[
   \int \frac{\tan^2(\theta)}{(1 + \tan^2(\theta))^2} \sec^2(\theta) \, d\theta
   \]

   Since \( 1 + \tan^2(\theta) = \sec^2(\theta) \), this simplifies to:

   \[
   \int \frac{\tan^2(\theta)}{\sec^4(\theta)} \sec^2(\theta) \, d\theta = \int \frac{\tan^2(\theta)}{\sec^2(\theta)} \, d\theta = \int \sin^2(\theta) \, d\theta
   \]

3. **Use the identity for \( \sin^2(\theta) \):**
   Recall the identity:
   \[
   \sin^2(\theta) = \frac{1 - \cos(2\theta)}{2}
   \]
   So, the integral becomes:
   \[
   \int \frac{1 - \cos(2\theta)}{2} \, d\theta
   \]

4. **Integrate:**
   The integral now breaks into two parts:
   \[
   \frac{1}{2} \int (1 - \cos(2\theta)) \, d\theta = \frac{1}{2} \left( \theta - \frac{\sin(2\theta)}{2} \right) + C
   \]
   Simplifying:
   \[
   \frac{1}{2} \theta - \frac{\sin(2\theta)}{4} + C
   \]

5. **Substitute back \( \theta \):**
   Since \( x = \tan(\theta) \), we have \( \theta = \tan^{-1}(x) \), and using the identity \( \sin(2\theta) = \frac{2\tan(\theta)}{1 + \tan^2(\theta)} = \frac{2x}{1 + x^2} \), the result becomes:

   \[
   \frac{1}{2} \tan^{-1}(x) - \frac{x}{2(1 + x^2)} + C
   \]

Thus, the final answer is:

\[
\boxed{\frac{1}{2} \tan^{-1}(x) - \frac{x}{2(1 + x^2)} + C}
\]

Would you like further clarifications, or have any questions?

### Related Questions:
1. How do you perform a trigonometric substitution in integration in general?
2. What other cases involve substituting \( x = \tan(\theta) \)?
3. How can you solve integrals involving powers of \( \tan(\theta) \)?
4. Why does the identity \( \sin^2(\theta) = \frac{1 - \cos(2\theta)}{2} \) help in integration?
5. What are other methods to solve the same integral without substitution?

### Tip:
When using trigonometric substitution, always simplify expressions using trigonometric identities before proceeding with integration.

Find the indefinite integral by using the substitution x = tan(θ): ∫(x² / (1 + x²)²) dx.

Learn how to find the indefinite integral using the trigonometric substitution x = tan(θ). This solution covers key mathematical concepts such as trigonometric identities and integration techniques. Suitable for advanced high school or college 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