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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.12
Capitolo 8, Problema 8.R.12

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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1
Identify the integral to solve: \(\int \frac{8x + 5}{2x^{2} + 3x + 1} \, dx\).
Factor the denominator if possible. The quadratic \(2x^{2} + 3x + 1\) factors as \((2x + 1)(x + 1)\).
Express the integrand as a sum of partial fractions: \(\frac{8x + 5}{(2x + 1)(x + 1)} = \frac{A}{2x + 1} + \frac{B}{x + 1}\), where \(A\) and \(B\) are constants to be determined.
Multiply both sides by the denominator \((2x + 1)(x + 1)\) to get: \(8x + 5 = A(x + 1) + B(2x + 1)\). Then, expand and collect like terms to form an equation in \(x\).
Solve the system of equations for \(A\) and \(B\) by equating coefficients of \(x\) and the constant terms. Once \(A\) and \(B\) are found, rewrite the integral as the sum of two simpler integrals and integrate each using the natural logarithm rule: \(\int \frac{1}{ax + b} \, dx = \frac{1}{a} \ln|ax + b| + C\).

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Partial Fraction Decomposition

Partial fraction decomposition is a technique used to break down a rational function into simpler fractions that are easier to integrate. It is especially useful when the denominator can be factored into linear or quadratic terms. This method transforms the integral into a sum of simpler integrals.
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Partial Fraction Decomposition: Distinct Linear Factors

Integration of Rational Functions

Integrating rational functions involves expressing the integrand as a ratio of polynomials and applying algebraic techniques like substitution or partial fractions. Recognizing the form of the integrand helps determine the appropriate method to simplify and integrate the expression.
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Intro to Rational Functions

Substitution Method

The substitution method simplifies integration by changing variables to transform the integral into a more manageable form. It is often used when the integrand contains a function and its derivative, allowing the integral to be rewritten in terms of a single variable.
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Euler's Method
Pratica correlata
Domanda del libro di testo

"Electric field due to a line of charge A total charge of Q is distributed uniformly on a line segment of length 2L along the y-axis (see figure). The x-component of the electric field at a point (a, 0) is given by

Eₓ(a) = (kQa/2L) ∫-L L dy/(a² + y²)^(3/2),

where k is a physical constant and a > 0.

a. Confirm that Eₓ(a)=kQ / a √(a²+L²)

b. Letting ρ=Q / 2 L be the charge density on the line segment, show that if L → ∞, then Eₓ(a) = 2kρ / a.

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Domanda del libro di testo

114. {Use of Tech} Arc length of the natural logarithm Consider the curve y = ln(x).

c. As a increases, L(a) increases as what power of a?

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Domanda del libro di testo

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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Domanda del libro di testo

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

43. ∫ eˣ sin(x) dx

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Domanda del libro di testo

92. Integral with a parameter For what values of p does the integral

∫ (from 1 to ∞) dx/xlnᵖ(x) converge, and what is its value (in terms of p)?

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108. Arc length Find the length of the curve y = ln(x) on the interval [1, e^2].

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