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Magazine Chapter 8: Problem 18
Converge. Evaluate the integrals without using tables. $$\int_{1}^{\infty} \frac{1}{x \sqrt{x^{2}-1}} d x$$
Short Answer
Expert verified
The integral converges to \( \frac{\pi}{2} \).
Step by step solution
01
Identify the Type of Integral
The given integral is an improper integral due to its upper limit of infinity. It requires evaluating the integral from 1 to infinity for the function \( \frac{1}{x \sqrt{x^2-1}} \).
02
Determine the Substitution
Use a trigonometric substitution to simplify the integral. Since the integrand has \( \sqrt{x^2-1} \), let's use the substitution \( x = \sec(\theta) \), so \( dx = \sec(\theta) \tan(\theta) d\theta \). Notice that \( \sqrt{x^2-1} = \tan(\theta) \).
03
Change the Limits of Integration
When \( x = 1 \), \( \sec(\theta) = 1 \), hence \( \theta = 0 \). As \( x \to \infty \), \( \theta \to \frac{\pi}{2} \). Change the limits of integration from \( x \)-values to \( \theta \)-values: \( 0 \) to \( \frac{\pi}{2} \).
04
Substitute and Simplify the Integral
Substitute \( x \) and \( dx \) into the integral: \[ \int_{0}^{\frac{\pi}{2}} \frac{1}{\sec(\theta) \cdot \tan(\theta)} \sec(\theta) \tan(\theta) d\theta = \int_{0}^{\frac{\pi}{2}} d\theta. \]The \( \sec(\theta) \) and \( \tan(\theta) \) terms cancel out.
05
Evaluate the Integral
The integral simplifies to \( \int_{0}^{\frac{\pi}{2}} d\theta \), which evaluates to \( \theta \Big|_{0}^{\frac{\pi}{2}} = \frac{\pi}{2} - 0 = \frac{\pi}{2}. \)
06
Conclusion on Convergence
The integral evaluates to a finite number, \( \frac{\pi}{2} \). Hence, the integral converges.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Trigonometric Substitution
Trigonometric substitution is a useful technique in calculus to simplify integrals involving square roots of quadratic expressions. In the given problem, we saw it used to tackle the integral
- The integral contains the term \( \sqrt{x^2-1} \), which resembles a trigonometric identity. Specifically, this is similar to the identity \( \tan^2(\theta) + 1 = \sec^2(\theta) \).
- By using the substitution \( x = \sec(\theta) \), we can express \( \sqrt{x^2-1} \) as \( \tan(\theta) \). This is due to the fact that \( \sec^2(\theta) - 1 = \tan^2(\theta) \).
- Our differential \( dx \) becomes \( \sec(\theta) \tan(\theta) d\theta \), due to the derivative of \( \sec(\theta) \).
Convergence of Integrals
Convergence examines whether an integral results in a finite value. Improper integrals, like the one we are given, where the upper limit is infinity, require careful analysis.
- An integral is said to converge if substituting boundaries results in a finite number.
- The integral from 1 to infinity of the function \( \frac{1}{x \sqrt{x^{2}-1}} \) initially appears complex but simplifies using trigonometric substitution.
- Once simplified, we found that the integral \( \int_{0}^{\frac{\pi}{2}} d\theta \) results in a finite value \( \frac{\pi}{2} \).
Calculus Techniques
Utilizing calculus techniques to solve integrals involves understanding various methods. Such methods make computations manageable when faced with complex expressions.
- Substitution Method: This is a powerful tool especially when dealing with trigonometric identities. It helps simplify difficult expressions by transforming variables.
- Recognizing Patterns: Identifying known trigonometric patterns within the integrands can direct us to appropriate substitutions, making calculations straightforward.
- Limit Evaluation: For improper integrals, evaluating limits at infinity is crucial. Transformations need altering of limits to fit the new variable, providing a solution.
