Skip to main content
Ch. 8 - Integration Techniques
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBriggs 3rd EditionCh. 8 - Integration TechniquesProblema 8.3.28
Capitolo 8, Problema 8.3.28

9–61. Trigonometric integrals Evaluate the following integrals.
28. ∫ 6 sec⁴x dx

Guida verificata passo dopo passo
1
Step 1: Recognize the integral ∫ 6 sec⁴x dx as a trigonometric integral involving secant raised to the fourth power. The goal is to simplify and evaluate this integral.
Step 2: Recall the standard formula for the integral of sec⁴x: ∫ sec⁴x dx = (1/3) tan³x + tan x + C. This formula can be applied directly to the given integral.
Step 3: Factor out the constant 6 from the integral to simplify the computation: ∫ 6 sec⁴x dx = 6 ∫ sec⁴x dx.
Step 4: Substitute the formula for ∫ sec⁴x dx into the expression: 6 ∫ sec⁴x dx = 6 [(1/3) tan³x + tan x + C].
Step 5: Simplify the expression by distributing the constant 6 across the terms inside the brackets: 6 [(1/3) tan³x + tan x + C] = 2 tan³x + 6 tan x + 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:
3m

Concetti chiave

Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.

Integration of Trigonometric Functions

Integration of trigonometric functions involves finding the antiderivative of functions that include trigonometric ratios. In this case, the integral of secant raised to a power, such as sec⁴x, requires knowledge of specific integration techniques and formulas related to trigonometric identities.
Video consigliato:
Percorso guidato
6:04
Introduction to Trigonometric Functions

Secant Function

The secant function, denoted as sec(x), is the reciprocal of the cosine function, expressed as sec(x) = 1/cos(x). Understanding the properties and behavior of the secant function is crucial for evaluating integrals involving secant, especially when raised to higher powers, as it can often be simplified using trigonometric identities.
Video consigliato:
Percorso guidato
6:22
Graphs of Secant and Cosecant Functions

Integration Techniques

Various techniques exist for integrating functions, including substitution, integration by parts, and recognizing patterns in integrals. For sec⁴x, one might use the identity sec²x = 1 + tan²x to rewrite the integral, making it easier to evaluate. Familiarity with these techniques is essential for solving complex integrals.
Video consigliato:
Percorso guidato
06:18
Integration by Parts for Definite Integrals
Pratica correlata
Domanda del libro di testo

42-47. Volumes of Solids Find the volume of the solid generated when the given region is revolved as described.

44. The region bounded by f(x) = sin(x) and the x-axis on [0, π] is revolved about the y-axis.

57
views
Domanda del libro di testo

7–64. Integration review Evaluate the following integrals.

10. ∫ e^(3 - 4x) dx

86
views
Domanda del libro di testo

69-72. Volumes of solids Find the volume of the following solids.

70. The region bounded by y = 1/[x²(x² + 2)²], y = 0, x = 1, and x = 2 is revolved about the y-axis.

48
views
Domanda del libro di testo

50-53. Reduction Formulas Use integration by parts to derive the following reduction formulas:

51. ∫ xⁿ cos(ax) dx = (xⁿ sin(ax))/a - (n/a) ∫ xⁿ⁻¹ sin(ax) dx, for a ≠ 0

93
views
Domanda del libro di testo

9–61. Trigonometric integrals Evaluate the following integrals.

37. ∫ [sec⁴(lnθ)]/θ dθ

62
views
Domanda del libro di testo

9–61. Trigonometric integrals Evaluate the following integrals.

22. ∫[π/4 to π/2] sin²(2x) cos³(2x) dx

78
views