교과서 질문
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
Ch. 8 - Integration Techniques
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
Not the one you use?Change textbookSelect a different textbook
Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
8장, 문제 8.3.28
9–61. Trigonometric integrals Evaluate the following integrals.
28. ∫ 6 sec⁴x dx
검증된 단계별 안내1
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.
비슷한 문제에 대한 검증된 영상 답변:
이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
영상 길이:
3m도움이 되었나요?
주요 개념
질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.
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.
추천 영상:
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.
추천 영상:
6:22Graphs 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.
추천 영상:
06:18Integration by Parts for Definite Integrals
관련 실천
교과서 질문
7–64. Integration review Evaluate the following integrals.
10. ∫ e^(3 - 4x) dx
86
views
교과서 질문
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
교과서 질문
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
교과서 질문
9–61. Trigonometric integrals Evaluate the following integrals.
37. ∫ [sec⁴(lnθ)]/θ dθ
62
views
교과서 질문
9–61. Trigonometric integrals Evaluate the following integrals.
22. ∫[π/4 to π/2] sin²(2x) cos³(2x) dx
78
views