교과서 질문
7–64. Integration review Evaluate the following integrals.
20. ∫ eˣ (1 + eˣ)⁹ (1 - eˣ) 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 8.7.24
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.
24. ∫ dt / √(1 + 4eᵗ)
검증된 단계별 안내1
Step 1: Recognize that the integral ∫ dt / √(1 + 4eᵗ) does not directly match a standard form in the table of integrals. Begin by identifying a substitution that simplifies the expression. Let u = 4eᵗ, which implies du = 4eᵗ dt or dt = du / (4eᵗ).
Step 2: Substitute u = 4eᵗ into the integral. The integral becomes ∫ (1 / √(1 + u)) * (du / 4eᵗ). Since eᵗ = u / 4, replace eᵗ with u / 4 in the denominator, simplifying the integral to ∫ (1 / √(1 + u)) * (du / u).
Step 3: Simplify the integral further. The expression now becomes (1/4) ∫ (1 / (u√(1 + u))) du. This matches a standard form in the table of integrals, which involves the substitution u = x² or another transformation to simplify the square root.
Step 4: Refer to the table of integrals for the standard form that matches ∫ (1 / (u√(1 + u))) du. Use the appropriate formula from the table to rewrite the integral in terms of u.
Step 5: After applying the formula from the table, back-substitute u = 4eᵗ to express the result in terms of t. Ensure the final expression is simplified and represents the indefinite integral.
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주요 개념
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Indefinite Integrals
Indefinite integrals represent a family of functions whose derivative is the integrand. They are expressed without limits and include a constant of integration. Understanding how to evaluate indefinite integrals is crucial for solving problems in calculus, as they provide the antiderivative of a function.
추천 영상:
05:04
Introduction to Indefinite Integrals
Completing the Square
Completing the square is a technique used to transform a quadratic expression into a perfect square trinomial. This method is often necessary for simplifying integrals involving quadratic terms, making them easier to integrate. It allows for the identification of standard integral forms in tables.
추천 영상:
05:22Completing the Square to Rewrite the Integrand
Integral Tables
Integral tables are collections of standard integrals that provide quick references for evaluating common integrals. They simplify the integration process by allowing students to look up known forms rather than calculating integrals from scratch. Familiarity with these tables is essential for efficiently solving complex integrals.
추천 영상:
08:09Tabular Integration by Parts
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