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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 8.6.85d
Chapter 8, Problem 8.6.85d

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.

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Step 1: Begin by analyzing the substitution u = tan(x). When using substitution in integration, we replace the original variable (x) with a new variable (u) and also compute the derivative of the substitution. For u = tan(x), the derivative is du/dx = sec²(x), or equivalently, dx = du / sec²(x).
Step 2: Rewrite the original integral ∫ (tan²(x) / (tan(x) - 1)) dx in terms of u. Since tan(x) = u, tan²(x) becomes u², and tan(x) - 1 becomes u - 1. Substitute these expressions into the integral.
Step 3: Replace dx with du / sec²(x) in the integral. The integral now becomes ∫ (u² / (u - 1)) * (1 / sec²(x)) du. At this point, sec²(x) must also be expressed in terms of u to complete the substitution.
Step 4: Recall the trigonometric identity sec²(x) = 1 + tan²(x). Since tan(x) = u, sec²(x) = 1 + u². Substitute this into the integral, replacing sec²(x) with 1 + u².
Step 5: Simplify the integral. After substituting sec²(x) = 1 + u², the integral becomes ∫ (u² / (u - 1)) * (1 / (1 + u²)) du. This does not simplify directly to ∫ (u² / (u - 1)) du, so the statement in the problem is false. The substitution leads to a more complex integral.

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Key Concepts

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

Substitution in Integration

Substitution is a technique used in integration to simplify the integrand by changing variables. By letting u = g(x), the integral can be transformed into a function of u, making it easier to evaluate. The differential dx is also converted using the derivative of g(x), which is essential for maintaining the equality of the integral.
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Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values of the variables. In this context, knowing the identity tan²(x) = sec²(x) - 1 can help simplify the integrand before applying substitution. Understanding these identities is crucial for manipulating expressions involving trigonometric functions effectively.
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Limits of Integration

When performing substitution in definite integrals, it is important to change the limits of integration to correspond with the new variable. This ensures that the area under the curve is accurately represented. In indefinite integrals, while limits are not a concern, understanding how they change with substitution is vital for definite integrals.
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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

82. A family of exponentials The curves y = x * e^(-a * x) are shown in the figure for a = 1, 2, and 3.

c. Find the area of the region bounded by y = x * e^(-a * x) and the x-axis on the interval [0, b]. Because this area depends on a and b, we call it A(a, b).

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Textbook Question

Gaussians An important function in statistics is the Gaussian (or normal distribution, or bell-shaped curve), f(x) = e^(-ax²).

c. Complete the square to evaluate ∫ from -∞ to ∞ of e^(-(ax² + bx + c)) dx, where a > 0, b, and c are real numbers.

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Textbook Question

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)

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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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