교과서 질문
Evaluate the integrals in Exercises 1–22.
∫ sin⁴(2x) cos(2x) dx
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Ch. 8 - Techniques of Integration
Hass15th EditionThomas' CalculusISBN: 9780137616077
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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
8장, 문제 8.1.32
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.
∫₋₁¹ (√(1 + x²) sin x) dx
검증된 단계별 안내1
First, examine the integrand \(\sqrt{1 + x^{2}} \sin x\) to determine if it has any symmetry properties. Note that \(\sqrt{1 + x^{2}}\) is an even function because replacing \(x\) by \(-x\) does not change its value, while \(\sin x\) is an odd function because \(\sin(-x) = -\sin x\).
Since the integrand is the product of an even function and an odd function, the overall integrand is an odd function. Recall that the product of an even function and an odd function is an odd function.
The integral of an odd function over a symmetric interval \([-a, a]\) is zero. This is because the areas on the negative and positive sides cancel each other out.
Therefore, without performing any algebraic manipulation or substitution, conclude that the value of the integral \(\int_{-1}^{1} \sqrt{1 + x^{2}} \sin x \, dx\) is zero due to the odd symmetry of the integrand.
If you want to verify this result, you could split the integral into two parts from \(-1\) to \(0\) and from \(0\) to \(1\) and check that they are negatives of each other, confirming the integral sums to zero.
비슷한 문제에 대한 검증된 영상 답변:
이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
영상 길이:
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주요 개념
질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.
Definite Integrals and Symmetry
Definite integrals calculate the net area under a curve between two limits. When the integrand involves symmetric intervals, analyzing whether the function is even, odd, or neither can simplify evaluation. For example, the integral of an odd function over [-a, a] is zero.
추천 영상:
가이드 코스
05:43
Definition of the Definite Integral
Properties of Even and Odd Functions
A function f(x) is even if f(-x) = f(x) and odd if f(-x) = -f(x). The product of an even and an odd function is odd, and the product of two even or two odd functions is even. Recognizing these properties helps determine integral values over symmetric intervals.
추천 영상:
가이드 코스
06:21Properties of Functions
Integration Techniques: Substitution and Algebraic Manipulation
When direct integration is difficult, substitution or algebraic manipulation can simplify the integral. Substitution changes variables to transform the integral into a more manageable form, while algebraic methods or trigonometric identities can rewrite the integrand for easier integration.
추천 영상:
07:01Integrals Involving Natural Logs: Substitution
관련 실천
교과서 질문
The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.
∫₀² dx / √|x − 1|
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교과서 질문
Evaluate the integrals in Exercises 33–52.
∫ eˣ sec³(eˣ) dx
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교과서 질문
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 -1 to 1 of (dθ / (θ² - 2θ))
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교과서 질문
Evaluate ∫ x³ √(1 - x²) dx using:
c. A trigonometric substitution.
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교과서 질문
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.
∫ (x² / (x² + 1)) dx
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