교과서 질문
59. Area of a segment of a circle
Use two approaches to show that the area of a cap (or segment) of a circle of radius r subtended by an angle θ (see figure) is given by:
A_seg = (1/2) r² (θ - sin θ)
b. Find the area using calculus.
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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 8.1.38
7–64. Integration review Evaluate the following integrals.
38. ∫ x / (x⁴ + 2x² + 1) dx
검증된 단계별 안내1
Step 1: Observe the integrand ∫ x / (x⁴ + 2x² + 1) dx. Notice that the denominator can be factored. Rewrite the denominator as (x² + 1)², since x⁴ + 2x² + 1 is a perfect square trinomial.
Step 2: Substitute u = x² + 1 to simplify the integral. Compute the derivative of u with respect to x: du/dx = 2x, which implies du = 2x dx.
Step 3: Rewrite the integral in terms of u. Substitute x dx with (1/2) du, and the denominator becomes u². The integral now becomes (1/2) ∫ 1/u² du.
Step 4: Apply the power rule for integration to ∫ 1/u² du. Recall that ∫ u⁻² du = -u⁻¹ + C, where C is the constant of integration.
Step 5: Substitute back u = x² + 1 into the result to express the solution in terms of x. The final answer will be in the form of -1/(x² + 1) + C.
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.
Integration Techniques
Integration techniques are methods used to find the integral of a function. Common techniques include substitution, integration by parts, and partial fraction decomposition. Understanding these methods is crucial for evaluating more complex integrals, such as the one presented in the question.
추천 영상:
06:18
Integration by Parts for Definite Integrals
Polynomial Functions
Polynomial functions are expressions that involve variables raised to whole number powers. In the integral ∫ x / (x⁴ + 2x² + 1) dx, the denominator is a polynomial of degree four. Recognizing the structure of polynomial functions helps in simplifying the integral and determining appropriate integration techniques.
추천 영상:
07:00Taylor Polynomials
Rational Functions
Rational functions are ratios of two polynomial functions. The integral in the question involves a rational function, which can often be simplified or decomposed for easier integration. Understanding how to manipulate rational functions is essential for effectively evaluating integrals like the one given.
추천 영상:
6:04Intro to Rational Functions
관련 실천
교과서 질문
23-64. Integration Evaluate the following integrals.
35. ∫ (x² + 12x - 4)/(x³ - 4x) dx
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교과서 질문
Evaluate the following integrals.
∫ x/(x² + 6x + 18) dx
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교과서 질문
65-76. Volumes Find the volume of the described solid of revolution or state that it does not exist.
75. The region bounded by f(x) = (4 - x)^(-1/3) and the x-axis on the interval [0, 4) is revolved about the y-axis.
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교과서 질문
7–40. Table look-up integrals Use a table of integrals to evaluate the following indefinite integrals. Some of the integrals require preliminary work, such as completing the square or changing variables, before they can be found in a table.
40. ∫ (e³ᵗ / √(4 + e²ᵗ)) dt
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교과서 질문
7–84. Evaluate the following integrals.
41. ∫ cot^(3/2)x · csc⁴x dx
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