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Ch. 8 - Integration Techniques
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBriggs 3rd EditionCh. 8 - Integration TechniquesProblema 8.R.25
Capitolo 8, Problema 8.R.25

2–74. Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals.
25. ∫ (from -3/2 to -1) dx/(4x² + 12x + 10)

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Step 1: Recognize that the denominator of the integrand, 4x² + 12x + 10, is a quadratic expression. To simplify the integral, complete the square for the quadratic expression in the denominator.
Step 2: Rewrite 4x² + 12x + 10 by factoring out 4 from the first two terms: 4(x² + 3x) + 10. Then complete the square inside the parentheses: x² + 3x = (x + 3/2)² - (3/2)². Substitute this back into the expression.
Step 3: After completing the square, the denominator becomes 4((x + 3/2)² - 9/4) + 10. Simplify this expression to get 4(x + 3/2)² + 1.
Step 4: Recognize that the integral now has the form ∫ dx / [a² + (x - h)²], which is a standard form for integration using the arctangent function. Use the formula ∫ dx / (a² + x²) = (1/a) * arctan(x/a) + C, adjusting for the shift in x.
Step 5: Apply the limits of integration (-3/2 to -1) to the result obtained from the arctangent formula. Evaluate the definite integral by substituting the limits into the antiderivative.

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Integration Techniques

Integration techniques are methods used to find the integral of a function. Common techniques include substitution, integration by parts, and partial fraction decomposition. Understanding these methods is essential for evaluating more complex integrals, especially those that cannot be solved using basic antiderivatives.
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Integration by Parts for Definite Integrals

Definite Integrals

A definite integral calculates the area under a curve between two specified limits. It is represented as ∫ from a to b f(x) dx, where 'a' and 'b' are the bounds of integration. The result of a definite integral is a numerical value that represents this area, which is crucial for applications in physics and engineering.
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Definition of the Definite Integral

Quadratic Functions

Quadratic functions are polynomial functions of the form ax² + bx + c, where a, b, and c are constants. In the context of integration, recognizing the structure of a quadratic expression can help in simplifying the integral, often by completing the square or using substitution to facilitate easier integration.
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Partial Fraction Decomposition: Irreducible Quadratic Factors
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122. Comparing areas The region R₁ is bounded by the graph of y = tan(x) and the x-axis on the interval [0, π/3].

The region R₂ is bounded by the graph of y = sec(x) and the x-axis on the interval [0, π/6]. Which region has the greater area?

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

86. ∫ (from -∞ to ∞) x³/(1 + x⁸) dx

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95–98. {Use of Tech} Numerical integration Estimate the following integrals using the Midpoint Rule M(n), the Trapezoidal Rule T(n), and Simpson’s Rule S(n) for the given values of n.

96. ∫ (from 1 to 3) dx/(x³ + x + 1); n = 4

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120. Equal volumes

a. Let R be the region bounded by the graph of f(x) = x^(-p) and the x-axis, for x ≥ 1. Let V₁ and V₂ be the volumes of the solids generated when R is revolved about the x-axis and the y-axis, respectively, if they exist. For what values of p (if any) is V₁ = V₂?

b. Repeat part (a) on the interval [0, 1].

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119. {Use of Tech} Comparing volumes Let R be the region bounded by y = ln(x), the x-axis, and the line x = a, where a > 1.

b. Find the volume V₂(a) of the solid generated when R is revolved about the y-axis (as a function of a).

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

57. ∫ (from 0 to √3/2) 4/(9 + 4x²) dx

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