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.
∫ (x^2 + 6x) / (x^2 + 3)^2 dx
33
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 8.3.44
Evaluate the integrals in Exercises 33–52.
∫ sec⁶(x) dx
Guida verificata passo dopo passo1
Recognize that the integral involves a high even power of secant: \(\int \sec^{6}(x) \, dx\). To simplify, express \(\sec^{6}(x)\) as \(\sec^{4}(x) \cdot \sec^{2}(x)\).
Rewrite \(\sec^{4}(x)\) as \((\sec^{2}(x))^{2}\), so the integral becomes \(\int (\sec^{2}(x))^{2} \cdot \sec^{2}(x) \, dx = \int \sec^{4}(x) \cdot \sec^{2}(x) \, dx\).
Use the identity \(\sec^{2}(x) = 1 + \tan^{2}(x)\) to express powers of secant in terms of tangent, or alternatively, use the reduction formula for powers of secant: \(\int \sec^{n}(x) \, dx = \frac{\sec^{n-2}(x) \tan(x)}{n-1} + \frac{n-2}{n-1} \int \sec^{n-2}(x) \, dx\) for \(n > 1\).
Apply the reduction formula with \(n=6\) to reduce the integral \(\int \sec^{6}(x) \, dx\) to an expression involving \(\int \sec^{4}(x) \, dx\).
Repeat the reduction process on \(\int \sec^{4}(x) \, dx\) until you reach integrals of \(\sec^{2}(x)\), which is straightforward to integrate, and then combine all parts to express the original integral.
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:
2mConcetti chiave
Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.
Integration of Powers of Secant
Integrating powers of secant functions often requires using reduction formulas or expressing secant in terms of tangent and secant to simplify the integral. For even powers, it is common to separate one sec²(x) factor and use trigonometric identities to reduce the power step-by-step.
Video consigliato:
05:42
Example 6: Integral of Secant & Cosecant
Trigonometric Identities
Key identities such as sec²(x) = 1 + tan²(x) help transform the integral into a more manageable form. These identities allow substitution and reduction of powers, making it easier to integrate complex trigonometric expressions.
Video consigliato:
7:17Verifying Trig Equations as Identities
Substitution Method
Substitution is often used when the integral contains a function and its derivative, such as tan(x) and sec²(x). By substituting u = tan(x), the integral can be rewritten in terms of u, simplifying the integration process.
Video consigliato:
07:33Euler's Method
Pratica correlata
Domanda del libro di testo
Evaluate the integrals in Exercises 1–24 using integration by parts.
∫ (r² + r + 1) e^r dr
32
views
Domanda del libro di testo
The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.
∫₀¹ (−ln(x)) dx
21
views
Domanda del libro di testo
Volume: Find the volume of the solid formed by revolving the region bounded by the graphs of y = sin x + sec x, y = 0, x = 0, and x = π/3 about the x-axis.
49
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.
∫ x^2 / √(x^2 - 4x + 5) dx
40
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 0 to ∞ of (dθ / (θ² - 1))
32
views