Textbook Question
7-56. Trigonometric substitutions Evaluate the following integrals using trigonometric substitution.
48. ∫ √(9 - 4x²) dx
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7. Antiderivatives & Indefinite Integrals
Indefinite Integrals
2:59 minutesProblem 8.4.68
Textbook Question
60–69. Completing the square Evaluate the following integrals.
68. ∫ dx / sqrt((x - 1)(3 - x))
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Completing the Square
Completing the square is a method used to transform a quadratic expression into a perfect square trinomial. This technique is essential for simplifying integrals involving square roots of quadratic expressions, allowing for easier integration. By rewriting the expression in the form (x - h)² - k, we can identify the vertex and facilitate the integration process.
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Trigonometric Substitution
Trigonometric substitution is a technique used in calculus to simplify integrals involving square roots. By substituting a variable with a trigonometric function, such as x = a sin(θ) or x = a cos(θ), we can transform the integral into a more manageable form. This method is particularly useful for integrals that involve expressions like √(a² - x²) or √(x² - a²).
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Definite and Indefinite Integrals
Integrals can be classified as definite or indefinite. An indefinite integral represents a family of functions and includes a constant of integration, while a definite integral calculates the area under a curve between two specific limits. Understanding the distinction is crucial for evaluating integrals correctly, especially when applying techniques like substitution or completing the square.
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