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Ch. 4 - Applications of Derivatives
Hass - Thomas' Calculus 15th Edition
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모든 교과서Hass 15th EditionCh. 4 - Applications of Derivatives문제 4.PE.83
4장, 문제 4.PE.83

Finding Indefinite Integrals
Find the indefinite integrals (most general antiderivatives) in Exercises 73–88. You may need to try a solution and then adjust your guess. Check your answers by differentiation.

∫ sec² s/10 ds

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Identify the integral to solve: \(\int \sec^{2}\left(\frac{s}{10}\right) \, ds\).
Recall the basic integral formula: \(\int \sec^{2}(x) \, dx = \tan(x) + C\), where \(C\) is the constant of integration.
Since the argument of the secant squared function is \(\frac{s}{10}\) instead of \(s\), use substitution to handle the inner function. Let \(u = \frac{s}{10}\).
Differentiate \(u\) with respect to \(s\) to find \(du\): \(du = \frac{1}{10} ds\), which implies \(ds = 10 \, du\).
Rewrite the integral in terms of \(u\): \(\int \sec^{2}(u) \, (10 \, du) = 10 \int \sec^{2}(u) \, du\). Then integrate using the basic formula to get \(10 \tan(u) + C\). Finally, substitute back \(u = \frac{s}{10}\) to express the answer in terms of \(s\).

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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 provides a family of functions whose derivative is the integrand.
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가이드 코스
05:04
Introduction to Indefinite Integrals

Integration of Trigonometric Functions

Certain trigonometric functions have standard integrals, such as ∫sec²(x) dx = tan(x) + C. Recognizing these forms helps in directly integrating expressions involving trigonometric functions.
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가이드 코스
6:04
Introduction to Trigonometric Functions

Substitution Method

When the integrand involves a function inside another function, substitution simplifies the integral by changing variables. For example, setting u = s/10 transforms the integral into a standard form easier to solve.
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07:33
Euler's Method
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교과서 질문

Initial Value Problems

Solve the initial value problems in Exercises 89–92.


dy/dx = (𝓍 + 1/𝓍)² , y(1)= 1

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교과서 질문

Initial Value Problems

Solve the initial value problems in Exercises 89–92.

d^3 r/dt^3 = - cos t; r''(0) = r'(0) = 0 , r(0) = -1

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교과서 질문

Applications


Suppose that f(x) = d/dx (1 − √x) and g(x) = d/dx (x + 2).


Find:


∫f(x) dx

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교과서 질문

Checking Antiderivative Formulas


Right, or wrong? Say which for each formula and give a brief reason for each answer.


∫√(2x + 1) dx = √(x² + x +C)

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교과서 질문

Finding Indefinite Integrals

Find the indefinite integrals (most general antiderivatives) in Exercises 73–88. You may need to try a solution and then adjust your guess. Check your answers by differentiation.

∫ 1/( r + 5)²dr

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교과서 질문

Finding Indefinite Integrals

Find the indefinite integrals (most general antiderivatives) in Exercises 73–88. You may need to try a solution and then adjust your guess. Check your answers by differentiation.

∫ (𝓍³ + 5𝓍 ―7) d𝓍

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