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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 8.9.101d
Chapter 8, Problem 8.9.101d

101. Many methods needed Show that the integral from ∫(from 0 to ∞)(sqrt(x) * ln x) / (1 + x)^2 dx equals pi, following these steps
d. Evaluate the remaining integral using the change of variables z = sqrt(x)

Verified step by step guidance
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Start with the integral after previous simplifications, which should be of the form \(\int_0^{\infty} \frac{\sqrt{x} \ln x}{(1+x)^2} \, dx\).
Apply the substitution \(z = \sqrt{x}\), which implies \(x = z^2\). Then, compute the differential \(dx\) in terms of \(dz\): \(dx = 2z \, dz\).
Rewrite the integral in terms of \(z\) by substituting \(x = z^2\), \(\sqrt{x} = z\), \(\ln x = \ln(z^2) = 2 \ln z\), and \(dx = 2z \, dz\). The integral becomes \(\int_0^{\infty} \frac{z \cdot 2 \ln z}{(1 + z^2)^2} \cdot 2z \, dz\).
Simplify the integrand by combining terms: \(z \cdot 2 \ln z \cdot 2z = 4 z^2 \ln z\), so the integral is \(\int_0^{\infty} \frac{4 z^2 \ln z}{(1 + z^2)^2} \, dz\).
Now, the integral is expressed in terms of \(z\) and can be evaluated using appropriate methods such as integration by parts or recognizing it as a standard integral form.

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Key Concepts

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

Change of Variables (Substitution) in Integration

This technique involves replacing the original variable with a new variable to simplify the integral. By setting z = sqrt(x), we rewrite the integral in terms of z, which often makes the integral easier to evaluate. It requires adjusting the differential dx accordingly and changing the limits of integration.
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Substitution With an Extra Variable

Improper Integrals over Infinite Intervals

Integrals with limits extending to infinity are called improper integrals. Evaluating them requires understanding limits and convergence. Here, the integral from 0 to ∞ must be handled carefully, ensuring the integral converges and applying appropriate techniques to evaluate it.
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Improper Integrals: Infinite Intervals

Logarithmic Functions within Integrals

Integrals involving logarithmic terms, such as ln(x), often require special attention due to their behavior near zero and infinity. Understanding properties of logarithms and how they interact with other functions in the integrand is essential for simplifying and evaluating the integral.
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Related Practice
Textbook Question

The Eiffel Tower Property Let R be the region between the curves y = e^(-c·x) and y = -e^(-c·x) on the interval [a, ∞), where a ≥ 0 and c > 0.

The center of mass of R is located at (x̄, 0), where x̄ = [∫(a to ∞) x·e^(-c·x) dx] / [∫(a to ∞) e^(-c·x) dx]

(The profile of the Eiffel Tower is modeled by these two exponential curves; see the Guided Project "The exponential Eiffel Tower")

d. Prove this property holds for any a ≥ 0 and c > 0:

The tangent lines to y = ±e^(-c·x) at x = a always intersect at R's center of mass

(Source: P. Weidman and I. Pinelis, Comptes Rendu, Mechanique, 332, 571-584, 2004)

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Textbook Question

66–71. {Use of Tech} Estimating error Refer to Theorem 8.1 in the following exercises.

66. Let f(x) = cos(x²).

d. Use Theorem 8.1 to find an upper bound on the absolute error in the estimate found in part (a).

Textbook Question

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

d. Using the substitution u = tan(x) in ∫ (tan²x / (tan x - 1)) dx leads to ∫ (u² / (u - 1)) du.

Textbook Question

2. Give an example of each of the following.

d. A repeated irreducible quadratic factor

Textbook Question

 Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

d. ∫(1/eˣ) dx = ln eˣ + C.

Textbook Question

77. Tabular integration Consider the integral ∫ f(x)g(x) dx, where f can be differentiated repeatedly and g can be integrated repeatedly

Let Gₖ represent the result of calculating k indefinite integrals of g (omitting constants of integration).

d. The tabular integration table from part (c) is easily extended to allow for as many steps as necessary in the process of integration by parts.

Evaluate ∫ x² e^(x/2) dx by constructing an appropriate table, and explain why the process terminates after four rows of the table have been filled in.

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