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Ch. 8 - Integration Techniques
Briggs - Calculus: Early Transcendentals 3rd Edition
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Tutti i libri di testoBriggs 3rd EditionCh. 8 - Integration TechniquesProblema 8.6.85d
Capitolo 8, Problema 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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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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Substitution With an Extra Variable

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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Verifying Trig Equations as Identities

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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One-Sided Limits
Pratica correlata
Domanda del libro di testo

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

d. The integral ∫ dx/(x² + 4x + 9) cannot be evaluated using a trigonometric substitution.

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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).

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

67. Let f(x) = √(x³ + 1).

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

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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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2. Give an example of each of the following.

d. A repeated irreducible quadratic factor

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