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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.10
Capitolo 8, Problema 8.R.10

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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Step 1: Simplify the integrand by performing polynomial long division. Divide \(x^3 + 3x^2 + 1\) by \(x^3 + 1\) to rewrite the integrand as a sum of simpler terms.
Step 2: After performing the division, express the integrand as \(Q(x) + \frac{R(x)}{x^3 + 1}\), where \(Q(x)\) is the quotient and \(R(x)\) is the remainder. This will simplify the integral into separate terms.
Step 3: Factorize \(x^3 + 1\) in the denominator of the fractional term. Recall that \(x^3 + 1 = (x + 1)(x^2 - x + 1)\). This will help in breaking down the fractional term further.
Step 4: Split the integral into separate parts: one for the polynomial quotient \(Q(x)\) and another for the fractional term \(\frac{R(x)}{(x + 1)(x^2 - x + 1)}\). Use partial fraction decomposition to handle the fractional term.
Step 5: Integrate each term separately. For the polynomial term \(Q(x)\), use basic power rule integration. For the fractional term, integrate using the results of the partial fraction decomposition, which may involve logarithmic and arctangent functions.

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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 crucial for evaluating complex integrals, as they allow for simplification and easier computation.
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Integration by Parts for Definite Integrals

Polynomial Long Division

Polynomial long division is a method used to divide one polynomial by another. In the context of integrals, it is often applied when the degree of the numerator is greater than or equal to the degree of the denominator. This technique simplifies the integrand into a more manageable form, making it easier to integrate.
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Rational Functions

A rational function is a ratio of two polynomials. When integrating rational functions, it is important to analyze their behavior and potential simplifications. Techniques such as partial fraction decomposition can be employed to break down complex rational functions into simpler components that are easier to integrate.
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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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"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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5-8. Compute the following estimates of ∫(0 to 8) f(x) dx using the graph in the figure.

6. T(4)

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29-34. {Use of Tech} Comparing the Midpoint and Trapezoid Rules

Apply the Midpoint and Trapezoid Rules to the following integrals. Make a table similar to Table 8.5 showing the approximations and errors for n = 4, 8, 16, and 32. The exact values of the integrals are given for computing the error.

30. ∫(0 to 6) (x³/16 - x) dx = 4

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

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