교과서 질문
66-68. Areas of regions (Use of Tech) Find the area of the following regions.
66. The region bounded by the curve y = (x - x²)/[(x + 1)(x² + 1)] and the x-axis from x = 0 to x = 1
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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 8.9.96
95–98. {Use of Tech} Numerical methods Use numerical methods or a calculator to approximate the following integrals as closely as possible. The exact value of each integral is given.
96. ∫(from 0 to ∞) (sin²x)/x² dx = π/2
검증된 단계별 안내1
Recognize that the integral is improper because the upper limit is infinity and the integrand involves a term \( \frac{\sin^2 x}{x^2} \), which is undefined at \( x = 0 \). So, we need to handle both the infinite limit and the behavior near zero carefully.
To approximate the integral numerically, first consider truncating the infinite upper limit to a large finite number \( M \), such that \( \int_0^\infty \frac{\sin^2 x}{x^2} dx \approx \int_0^M \frac{\sin^2 x}{x^2} dx \). Choose \( M \) large enough to capture the significant area under the curve.
Next, address the integrand's behavior near \( x = 0 \). Since \( \frac{\sin^2 x}{x^2} \) is indeterminate at zero, use the limit \( \lim_{x \to 0} \frac{\sin^2 x}{x^2} = 1 \) to define the value at zero or use a small positive number \( \epsilon \) as the lower limit instead of zero.
Apply a numerical integration method such as Simpson's rule, trapezoidal rule, or use a numerical integration function on a calculator or software to approximate \( \int_\epsilon^M \frac{\sin^2 x}{x^2} dx \). Make sure to use sufficiently small step sizes to improve accuracy.
Compare your numerical approximation to the exact value \( \frac{\pi}{2} \) to check the accuracy of your method. Adjust \( M \) and the step size if necessary to get a closer approximation.
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.
Improper Integrals
Improper integrals involve integration over an infinite interval or integrands with unbounded behavior. In this problem, the integral extends from 0 to infinity, requiring techniques to handle infinite limits or singularities to evaluate or approximate the integral.
추천 영상:
11:11
Improper Integrals: Infinite Intervals
Numerical Integration Methods
Numerical integration methods approximate definite integrals when exact solutions are difficult. Techniques like Simpson's rule, trapezoidal rule, or adaptive quadrature can be used, often with computational tools, to estimate the value of integrals, especially those with infinite limits.
추천 영상:
07:33Euler's Method
Behavior of the Integrand and Convergence
Understanding the integrand's behavior, such as oscillations and decay, is crucial for convergence of improper integrals. Here, (sin²x)/x² oscillates but decays sufficiently fast, ensuring the integral converges to a finite value, which justifies numerical approximation methods.
추천 영상:
07:51Choosing a Convergence Test
관련 실천
교과서 질문
9–61. Trigonometric integrals Evaluate the following integrals.
34. ∫ tan⁹x sec⁴x dx
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교과서 질문
27. {Use of Tech} Midpoint Rule, Trapezoid Rule, and relative error
Find the Midpoint and Trapezoid Rule approximations to ∫(0 to 1) sin(πx) dx using n = 25 subintervals. Compute the relative error of each approximation.
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교과서 질문
87-92. An integrand with trigonometric functions in the numerator and denominator can often be converted to a rational function using the substitution u = tan(x/2) or, equivalently, x = 2 tan⁻¹u. The following relations are used in making this change of variables.
A: dx = 2/(1 + u²) du
B: sin x = 2u/(1 + u²)
C: cos x = (1 - u²)/(1 + u²)
91. Evaluate ∫[0 to π/2] dθ/(cos θ + sin θ).
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교과서 질문
9–61. Trigonometric integrals Evaluate the following integrals.
25. ∫ sin²x cos⁴x dx
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교과서 질문
64. Using a computer algebra system, it was determined that
∫x(x+1)^8 dx = (x^10)/10 + (8x^9)/9 + (7x^8)/2 + 8x^7 + (35x^6)/3 + (56x^5)/5 + 7x^4 + (8x^3)/3 + x^2/2 + C.
Use integration by substitution to evaluate ∫x(x+1)^8 dx.
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