Domanda del libro di testo
The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.
∫₀² (s + 1) / √(4 − s²) ds
29
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 8.1.30
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.
∫ 3 sinh(x/2 + ln 5) dx
Guida verificata passo dopo passo1
Recognize that the integral involves the hyperbolic sine function \( \sinh \left( \frac{x}{2} + \ln 5 \right) \). Recall that \( \sinh(u) = \frac{e^{u} - e^{-u}}{2} \), which can be useful for integration.
Let \( u = \frac{x}{2} + \ln 5 \). Then, compute \( du = \frac{1}{2} dx \), which implies \( dx = 2 du \). This substitution will simplify the integral.
Rewrite the integral in terms of \( u \): \( \int 3 \sinh(u) dx = \int 3 \sinh(u) \cdot 2 du = \int 6 \sinh(u) du \).
Integrate \( \sinh(u) \) with respect to \( u \). Recall that \( \int \sinh(u) du = \cosh(u) + C \). So, \( \int 6 \sinh(u) du = 6 \cosh(u) + C \).
Substitute back \( u = \frac{x}{2} + \ln 5 \) to express the answer in terms of \( x \): the integral is \( 6 \cosh \left( \frac{x}{2} + \ln 5 \right) + C \).
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:
1mConcetti chiave
Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.
Hyperbolic Functions
Hyperbolic functions, such as sinh(x), are analogs of trigonometric functions but based on hyperbolas. The sinh function is defined as (e^x - e^(-x))/2 and has properties useful for integration, including linearity and known derivatives.
Video consigliato:
Percorso guidato
5:50
Asymptotes of Hyperbolas
Integration by Substitution
Integration by substitution involves changing variables to simplify an integral. By letting u equal an inner function (e.g., u = x/2 + ln 5), the integral can be rewritten in terms of u, making it easier to evaluate.
Video consigliato:
04:27Substitution With an Extra Variable
Linearity of Integration
The linearity property states that constants can be factored out of integrals and sums of functions can be integrated term-by-term. This allows the constant 3 to be factored out, simplifying the integral before applying substitution.
Video consigliato:
Percorso guidato
07:17Linearization
Pratica correlata
Domanda del libro di testo
Use any method to evaluate the integrals in Exercises 15–38. Most will require trigonometric substitutions, but some can be evaluated by other methods.
∫ (x² dx) / (x² - 1)^(5/2), where x > 1
16
views
Domanda del libro di testo
Evaluate the integrals in Exercises 31–56. Some integrals do not require integration by parts.
∫ x² sin(x³) dx
19
views
Domanda del libro di testo
Use integration by parts to obtain the formula ∫ √(1 - x²) dx = (1/2) x √(1 - x²) + (1/2) ∫ 1 / √(1 - x²) dx.
23
views
Domanda del libro di testo
The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.
∫₋∞² (2 dx) / (x² + 4)
21
views
Domanda del libro di testo
The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.
∫₋₈¹ dx / x^(1/3)
26
views