Domanda del libro di testo
Evaluate the integrals in Exercises 1–14.
∫ (2 dx) / (x³ √(x² - 1)), where x > 1
20
views
Ch. 8 - Techniques of Integration
Hass15th EditionThomas' CalculusISBN: 9780137616077
Non è quello che usi tu?Cambia libro di testoScegli un libro di testo diverso
TOC.tableOfContents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
Capitolo 8, Problema 8.3.34
Evaluate the integrals in Exercises 33–52.
∫ sec(x) tan²(x) dx
Guida verificata passo dopo passo1
Recall the trigonometric identity: \(\tan^2(x) = \sec^2(x) - 1\). This allows us to rewrite the integral in terms of secant functions.
Rewrite the integral as \(\int \sec(x) \tan^2(x) \, dx = \int \sec(x) (\sec^2(x) - 1) \, dx\).
Distribute \(\sec(x)\) inside the integral to get \(\int (\sec^3(x) - \sec(x)) \, dx\).
Split the integral into two separate integrals: \(\int \sec^3(x) \, dx - \int \sec(x) \, dx\).
Use known integration techniques: recall that \(\int \sec(x) \, dx\) is a standard integral, and for \(\int \sec^3(x) \, dx\), use integration by parts or a reduction formula to express it in terms of \(\int \sec(x) \, dx\).
Risposta video verificata per un problema simile:
Questa soluzione video è stata consigliata dai nostri tutor come utile per risolvere questo problema.
Durata del video:
4mConcetti chiave
Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.
Trigonometric Identities
Trigonometric identities like tan²(x) = sec²(x) - 1 help simplify integrals involving trigonometric functions. Recognizing and applying these identities can transform complex expressions into more manageable forms for integration.
Video consigliato:
7:17
Verifying Trig Equations as Identities
Integration Techniques for Trigonometric Functions
Integrating functions involving sec(x) and tan(x) often requires substitution or rewriting the integrand using identities. Familiarity with standard integrals such as ∫ sec(x) dx and ∫ sec(x) tan(x) dx is essential for solving these problems efficiently.
Video consigliato:
Percorso guidato
6:04Introduction to Trigonometric Functions
Substitution Method
The substitution method involves choosing a part of the integrand as a new variable to simplify the integral. For example, setting u = sec(x) or u = tan(x) can reduce the integral to a basic form, making it easier to evaluate.
Video consigliato:
07:33Euler's Method
Pratica correlata
Domanda del libro di testo
Evaluate the integrals in Exercises 39–54.
∫ (e⁴t + 2e²t - e^t) / (e²t + 1) dt
5
views
Domanda del libro di testo
The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.
∫₀^∞ 2e^(−θ) sinθ dθ
17
views
Domanda del libro di testo
In Exercises 35–68, use integration, the Direct Comparison Test, or the Limit Comparison Test to test the integrals for convergence. If more than one method applies, use whatever method you prefer.
∫ from -∞ to ∞ of ((dx) / (e^x + e^(-x)))
46
views
Domanda del libro di testo
The integrals in Exercises 1–44 are in no particular order. Evaluate each integral using any algebraic method, trigonometric identity, or substitution you think is appropriate.
∫ (6 dy / √y(1 + y))
27
views
Domanda del libro di testo
Evaluate the integrals in Exercises 1–14.
∫ √(1 - 9t²) dt
37
views