### Problem Statement:**Evaluate** \[ I = \int \frac{1}{x^2 \sqrt{9 + 4x^2}} \, dx \]### Detailed Explanation:To solve this integral, you'll often use substitution methods or trigonometric identities. This type of integral is usually found in calculus courses and is particularly relevant in problems involving trigonometric integrals, complex integrals, or integrals containing radicals.To guide you through the process, here's a step-by-step method using trigonometric substitution, one of the most systematic approaches for solving integrals with square roots. For this particular integral, we will consider an appropriate trigonometric substitution to simplify the integrand before integrating.1. **Substitution**: Let $ x = \frac{3}{2} \tan(\theta) $. Then $ dx = \frac{3}{2} \sec^2(\theta) \, d\theta $.2. **Simplifying the Integral**: Substitute $ x = \frac{3}{2} \tan(\theta) $ and $ dx = \frac{3}{2} \sec^2(\theta) \, d\theta $ into the integral. 3. **Solve the Resulting Trigonometric Integral**: Simplify the expression inside the integral to eventually integrate with respect to $ \theta $.4. **Back-Substitution**: Convert back from $ \theta $ to $ x $ after integrating.By following these steps, you will be able to evaluate the integral $ I $. This concept will be useful and often appears in higher mathematics and engineering problems, where the understanding of integral techniques is essential.### Conclusion:The detailed solution to the integral involves recognizing the structure of the integrand and applying trigonometric substitutions appropriately. This methodology showcases integral calculus's flexibility and widespread application. For further practice, try similar integrals or explore other substitution methods and check your results using software tools or integrating with different techniques to solidify your understanding.

Answered: Image /qna-images/answer/383d8dcd-d20e-44c1-a3bc-c4d6fe7441b5.jpg

Answered: 1 Evaluate I = dx x²V9+ 4x²

Math

Calculus

1 Evaluate I = dx x²V9+ 4x²

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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Equations and Inequations

Equations and inequalities describe the relationship between two mathematical expressions.

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A linear function can just be a constant, or it can be the constant multiplied with the variable like x or y. If the variables are of the form, x2, x1/2 or y2 it is not linear. The exponent over the variables should always be 1.

Question

### Problem Statement:

**Evaluate**

\[ I = \int \frac{1}{x^2 \sqrt{9 + 4x^2}} \, dx \]

### Detailed Explanation:

To solve this integral, you'll often use substitution methods or trigonometric identities. This type of integral is usually found in calculus courses and is particularly relevant in problems involving trigonometric integrals, complex integrals, or integrals containing radicals.

To guide you through the process, here's a step-by-step method using trigonometric substitution, one of the most systematic approaches for solving integrals with square roots. For this particular integral, we will consider an appropriate trigonometric substitution to simplify the integrand before integrating.

1. **Substitution**:
Let $ x = \frac{3}{2} \tan(\theta) $. Then $ dx = \frac{3}{2} \sec^2(\theta) \, d\theta $.

2. **Simplifying the Integral**:
Substitute $ x = \frac{3}{2} \tan(\theta) $ and $ dx = \frac{3}{2} \sec^2(\theta) \, d\theta $ into the integral.

3. **Solve the Resulting Trigonometric Integral**:
Simplify the expression inside the integral to eventually integrate with respect to $ \theta $.

4. **Back-Substitution**:
Convert back from $ \theta $ to $ x $ after integrating.

By following these steps, you will be able to evaluate the integral $ I $. This concept will be useful and often appears in higher mathematics and engineering problems, where the understanding of integral techniques is essential.

### Conclusion:

The detailed solution to the integral involves recognizing the structure of the integrand and applying trigonometric substitutions appropriately. This methodology showcases integral calculus's flexibility and widespread application.

For further practice, try similar integrals or explore other substitution methods and check your results using software tools or integrating with different techniques to solidify your understanding.

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