교과서 질문
Theory and Examples
Maximum height of a vertically moving body The height of a body moving vertically is given by s = −12gt² + υ₀t + s₀, g > 0, with s in meters and t in seconds. Find the body’s maximum height.
209
views
Ch. 4 - Applications of Derivatives
Hass15th EditionThomas' CalculusISBN: 9780137616077
당신이 사용하는 게 아니라요?교과서 변경다른 교과서를 선택하세요
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
4장, 문제 4.7.39
Finding Indefinite Integrals
In Exercises 17–56, find the most general antiderivative or indefinite integral. You may need to try a solution and then adjust your guess. Check your answers by differentiation.
∫(−3csc²x)dx
검증된 단계별 안내1
Recognize that the integral is of the form \(\int -3 \csc^{2}x \, dx\), where \(-3\) is a constant multiplier and \(\csc^{2}x\) is a standard trigonometric function whose integral is known.
Recall the basic integral formula: \(\int \csc^{2}x \, dx = -\cot x + C\), where \(C\) is the constant of integration.
Use the constant multiple rule for integrals, which allows you to factor out constants: \(\int -3 \csc^{2}x \, dx = -3 \int \csc^{2}x \, dx\).
Substitute the known integral result into the expression: \(-3 \int \csc^{2}x \, dx = -3 (-\cot x + C) = 3 \cot x + C'\), where \(C'\) is a new constant of integration.
Verify your result by differentiating \(3 \cot x + C'\) and confirming that it equals the original integrand \(-3 \csc^{2}x\).
비슷한 문제에 대한 검증된 영상 답변:
이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
영상 길이:
59s도움이 되었나요?
주요 개념
질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.
Indefinite Integral
An indefinite integral represents the most general antiderivative of a function, expressed with a constant of integration (C). It reverses differentiation and includes all possible functions whose derivative matches the integrand.
추천 영상:
가이드 코스
05:04
Introduction to Indefinite Integrals
Integration of Trigonometric Functions
Integrating trigonometric functions like csc²x involves knowing standard integral formulas, such as ∫csc²x dx = -cot x + C. Recognizing these forms simplifies finding antiderivatives.
추천 영상:
가이드 코스
6:04Introduction to Trigonometric Functions
Verification by Differentiation
After finding an indefinite integral, differentiating the result should return the original integrand. This step confirms the correctness of the antiderivative and helps identify any errors.
추천 영상:
가이드 코스
05:53Finding Differentials
관련 실천
교과서 질문
[Technology Exercises] When solving Exercises 14–30, you may need to use appropriate technology (such as a calculator or a computer).
26. Factoring a quartic Find the approximate values of r_1 through r_4 in the factorization
8x^4-14x^3-9x^2+11x-1=8(x-r_1)(x-r_2)(x-r_3)(x-r_4)
207
views
교과서 질문
Finding Critical Points
In Exercises 41–50, determine all critical points and all domain endpoints for each function.
y = x − 3x²ᐟ³
193
views
교과서 질문
Business and Economics
62. Production level Suppose that c(x)=x^3-20x^2 + 20,000x is the cost of manufacturing x items. Find a production level that will minimize the average cost of making x items.
182
views
교과서 질문
Identifying Extrema
In Exercises 19–40:
a. Find the open intervals on which the function is increasing and those on which it is decreasing.
b. Identify the function’s local extreme values, if any, saying where they occur.
g(x) = x√8 − x²
135
views
교과서 질문
Initial Value Problems
Solve the initial value problems in Exercises 71–90.
d³y/dx³ = 6; y″(0) = −8, y′(0) = 0, y(0) = 5
14
views