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Ch. 8 - Integration Techniques
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 8 - Integration TechniquesProblem 8.1.12
Chapter 8, Problem 8.1.12

7–64. Integration review Evaluate the following integrals.
12. ∫ from -5 to 0 of dx / √(4 - x)

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Step 1: Recognize the integral ∫ dx / √(4 - x) as a definite integral with limits of integration from -5 to 0. The integrand involves a square root, which suggests a substitution method might be helpful.
Step 2: Let u = 4 - x. Then, differentiate u with respect to x to find du/dx = -1, or equivalently, dx = -du. Substitute u into the integral to replace x and dx.
Step 3: Adjust the limits of integration according to the substitution. When x = -5, u = 4 - (-5) = 9. When x = 0, u = 4 - 0 = 4. The integral now becomes ∫ from u=9 to u=4 of -du / √u.
Step 4: Simplify the integral using the negative sign and rewrite it as -∫ from u=9 to u=4 of du / √u. Recognize that 1/√u can be expressed as u^(-1/2), which is a standard power rule for integration.
Step 5: Apply the power rule for integration to u^(-1/2). The integral of u^(-1/2) is 2√u. Evaluate this antiderivative at the new limits of integration (u=9 and u=4), and subtract the results to find the value of the definite integral.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Definite Integrals

A definite integral represents the signed area under a curve between two specified limits. It is denoted as ∫ from a to b of f(x) dx, where 'a' and 'b' are the lower and upper limits, respectively. The result of a definite integral is a numerical value that quantifies the accumulation of the function's values over the interval [a, b].
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Definition of the Definite Integral

Integration Techniques

Integration techniques are methods used to find the integral of a function. Common techniques include substitution, integration by parts, and recognizing standard forms. For the integral ∫ dx / √(4 - x), a substitution method can simplify the expression, making it easier to evaluate the integral.
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Square Root Functions

Square root functions, such as √(4 - x), are important in calculus as they often appear in integrals and derivatives. Understanding their behavior, including their domain and range, is crucial for evaluating integrals involving them. In this case, the expression under the square root must be non-negative, which influences the limits of integration.
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Related Practice
Textbook Question

11-14. {Use of Tech} Compute the absolute and relative errors in using c to approximate x.

12. x = √2; c = 1.414

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2–74. Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals.

12. ∫ (8x + 5)/(2x² + 3x + 1) dx

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Textbook Question

9–40. Integration by parts Evaluate the following integrals using integration by parts.

14. ∫ s · e⁻²ˢ ds

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60–69. Completing the square Evaluate the following integrals.

62. ∫ du / (2u² - 12u + 36)

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2–74. Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals.

10. ∫ (x³ + 3x² + 1)/(x³ + 1) dx

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82-88. Improper integrals Evaluate the following integrals or show that the integral diverges.

82. ∫ (from -∞ to -1) dx/(x - 1)⁴

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