교과서 질문
7–58. Improper integrals Evaluate the following integrals or state that they diverge.
10. ∫ (from 0 to ∞) e⁻²ˣ dx
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Ch. 8 - Integration Techniques
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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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 8.6.56
7–84. Evaluate the following integrals.
56. ∫ from π to 3π/2 sin2x e^(sin²x) dx
검증된 단계별 안내1
Step 1: Recognize that the integral involves a composition of functions, specifically sin²(x) and e^(sin²(x)). This suggests that substitution might be a useful technique to simplify the integral.
Step 2: Let u = sin²(x). Then, compute the derivative of u with respect to x: du/dx = 2sin(x)cos(x). This implies that du = 2sin(x)cos(x)dx.
Step 3: Rewrite the integral in terms of u. Notice that sin²(x) is replaced by u, and the differential dx is replaced by du/(2sin(x)cos(x)). The integral becomes ∫ e^u du/2.
Step 4: Adjust the limits of integration. When x = π, sin²(x) = sin²(π) = 0. When x = 3π/2, sin²(x) = sin²(3π/2) = 1. Thus, the new limits for u are from 0 to 1.
Step 5: Integrate ∫ e^u du/2 over the interval [0, 1]. This involves finding the antiderivative of e^u, which is e^u, and then evaluating it at the new limits of integration. Multiply the result by 1/2 to account for the factor outside the integral.
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
A definite integral represents the signed area under a curve between two specified limits. In this case, the integral is evaluated from π to 3π/2, which means we are calculating the area under the curve of the function sin²x e^(sin²x) within that interval. Understanding how to set up and evaluate definite integrals is crucial for solving the problem.
추천 영상:
05:43
Definition of the Definite Integral
Integration Techniques
Integration techniques are methods used to find the integral of a function. Common techniques include substitution, integration by parts, and recognizing patterns in integrals. For the given integral, recognizing that a substitution involving sin²x could simplify the expression is essential for finding the solution efficiently.
추천 영상:
06:18Integration by Parts for Definite Integrals
Exponential Functions
Exponential functions, such as e^(sin²x), are functions where the variable is in the exponent. They often arise in calculus problems involving growth and decay, and their properties can significantly affect the behavior of integrals. Understanding how to manipulate and integrate exponential functions is vital for evaluating the given integral.
추천 영상:
6:13Exponential Functions
관련 실천
교과서 질문
79–82. {Use of Tech} Double table look-up The following integrals may require more than one table look-up. Evaluate the integrals using a table of integrals, and then check your answer with a computer algebra system.
79. ∫ x sin⁻¹(2x) dx
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교과서 질문
7–64. Integration review Evaluate the following integrals.
32. ∫ from 0 to 2 of x / (x² + 4x + 8) dx
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교과서 질문
1. State the half-angle identities used to integrate sin²x and cos²x.
173
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교과서 질문
9–61. Trigonometric integrals Evaluate the following integrals.
59. ∫ from 0 to π/2 of √(1 - cos2x) dx
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교과서 질문
70. Different methods Let I=∫(x+2)/(x+4)dx.
b. Evaluate I without performing long division on the integrand.
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