Domanda del libro di testo
In Exercises 69–80, determine whether the improper integral converges or diverges. If it converges, evaluate the integral.
∫₋∞⁰ x² e^(x³) dx
14
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 8.4.2
Evaluate the integrals in Exercises 1–14.
∫ (3 dx) / √(1 + 9x²)
Guida verificata passo dopo passo1
Recognize that the integral has the form \(\int \frac{3 \, dx}{\sqrt{1 + 9x^2}}\), which resembles the standard integral \(\int \frac{dx}{\sqrt{a^2 + x^2}}\) whose antiderivative involves a hyperbolic or inverse hyperbolic function or a logarithm depending on the substitution.
Identify the constant inside the square root: here, \(a^2 = 1\) and the term with \(x^2\) is \(9x^2 = (3x)^2\). This suggests using a substitution to simplify the expression under the square root.
Make the substitution \(u = 3x\), which implies \(du = 3 \, dx\) or equivalently \(dx = \frac{du}{3}\). This will help rewrite the integral in terms of \(u\).
Rewrite the integral in terms of \(u\): substitute \(3 \, dx = du\) and \(\sqrt{1 + 9x^2} = \sqrt{1 + u^2}\), so the integral becomes \(\int \frac{du}{\sqrt{1 + u^2}}\).
Recall the antiderivative formula \(\int \frac{du}{\sqrt{1 + u^2}} = \sinh^{-1}(u) + C\) or equivalently \(\ln|u + \sqrt{1 + u^2}| + C\). After integrating, substitute back \(u = 3x\) to express the answer in terms of \(x\).
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:
11mConcetti chiave
Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.
Integration of Functions Involving Square Roots
This concept involves integrating functions where the integrand contains a square root expression, often requiring substitution or recognition of standard integral forms. Understanding how to manipulate and simplify the integrand is crucial for solving such integrals.
Video consigliato:
07:01
Integrals Involving Natural Logs: Substitution
Trigonometric Substitution
Trigonometric substitution is a technique used to simplify integrals involving expressions like √(a² + x²), √(a² - x²), or √(x² - a²). By substituting x with a trigonometric function, the integral becomes easier to evaluate using trigonometric identities.
Video consigliato:
Percorso guidato
6:04Introduction to Trigonometric Functions
Standard Integral Formulas
Familiarity with standard integral formulas, such as ∫ dx / √(a² + x²) = ln|x + √(x² + a²)| + C, helps quickly evaluate integrals without lengthy calculations. Recognizing these forms allows for efficient problem-solving.
Video consigliato:
Percorso guidato
3:40Circles in Standard Form Example 1
Pratica correlata
Domanda del libro di testo
The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.
∫₂⁴ dt / [t√(t² − 4)]
27
views
Domanda del libro di testo
Evaluate the integrals in Exercises 1–24 using integration by parts.
∫ t² e^(4t) dt
42
views
Domanda del libro di testo
In Exercises 27–40, use a substitution to change the integral into one you can find in the table. Then evaluate the integral.
∫ dt / (tan(t)√4 - sin^2(t))
21
views
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.
∫ dx / (1 + x²)
19
views
Domanda del libro di testo
In Exercises 39–48, use an appropriate substitution and then a trigonometric substitution to evaluate the integrals.
∫ (e^{t} dt) / ((1 + e^{2t})^{3/2}) from ln(3/4) to ln(4/3)
37
views