Textbook Question
23–68. Indefinite integrals Determine the following indefinite integrals. Check your work by differentiation.
∫ (1/2y)dy
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7. Antiderivatives & Indefinite Integrals
Indefinite Integrals
3:06 minutesProblem 4.9.55
Textbook Question
23–68. Indefinite integrals Determine the following indefinite integrals. Check your work by differentiation.
∫ (4/x√(x² - 1))dx
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Rewrite the integral in a more convenient form: \( \int \frac{4}{x\sqrt{x^2 - 1}} \, dx \). Notice that the integrand suggests a substitution involving a trigonometric identity.
Use the substitution \( x = \sec(\theta) \), which implies \( dx = \sec(\theta)\tan(\theta) \, d\theta \) and \( \sqrt{x^2 - 1} = \tan(\theta) \). Substitute these into the integral.
After substitution, the integral becomes \( \int \frac{4}{\sec(\theta) \cdot \tan(\theta)} \cdot \sec(\theta)\tan(\theta) \, d\theta \). Simplify the expression by canceling terms.
The simplified integral is \( \int 4 \, d\theta \), which is straightforward to integrate. Perform the integration to get \( 4\theta + C \), where \( C \) is the constant of integration.
Finally, reverse the substitution \( x = \sec(\theta) \) to express \( \theta \) in terms of \( x \). Since \( \theta = \sec^{-1}(x) \), the result becomes \( 4\sec^{-1}(x) + C \). Verify the solution by differentiating it to ensure it matches the original integrand.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Indefinite Integrals
Indefinite integrals represent a family of functions whose derivative is the integrand. They are expressed without limits and include a constant of integration, typically denoted as 'C'. The process of finding an indefinite integral is known as integration, which is the reverse operation of differentiation.
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Integration Techniques
Various techniques are used to solve integrals, including substitution, integration by parts, and trigonometric identities. For the integral ∫ (4/x√(x² - 1))dx, recognizing the form of the integrand can suggest a suitable method, such as trigonometric substitution, to simplify the integration process.
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Verification by Differentiation
After finding an indefinite integral, it is essential to verify the result by differentiating the obtained function. This process ensures that the derivative of the integrated function returns to the original integrand, confirming the correctness of the integration. This step is crucial for validating the solution in calculus.
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