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Ch. 4 - Applications of Derivatives
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077당신이 사용하는 게 아니라요?교과서 변경
모든 교과서Hass 15th EditionCh. 4 - Applications of Derivatives문제 4.7.54
4장, 문제 4.7.54

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.


∫(1 − cot²x) dx

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Recall the trigonometric identity involving cotangent and cosecant: \(\cot^{2}x + 1 = \csc^{2}x\). From this, express \(1 - \cot^{2}x\) in terms of \(\csc^{2}x\).
Rewrite the integrand using the identity: \(1 - \cot^{2}x = \csc^{2}x\).
Recognize that the integral now becomes \(\int \csc^{2}x \, dx\), which is a standard integral in calculus.
Recall the antiderivative of \(\csc^{2}x\) is \(-\cot x + C\), where \(C\) is the constant of integration.
Write the general solution as \(-\cot x + C\) and verify by differentiating \(-\cot x\) to ensure it returns the original integrand \(1 - \cot^{2}x\).

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주요 개념

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Indefinite Integral and Antiderivative

An indefinite integral represents the most general antiderivative of a function, including an arbitrary constant C. It reverses differentiation, finding a function whose derivative matches the integrand. Understanding this helps in solving integrals without specified limits.
추천 영상:
05:04
Introduction to Indefinite Integrals

Trigonometric Identities

Trigonometric identities, such as 1 + cot²x = csc²x, simplify integrals involving trigonometric functions. Recognizing and applying these identities transforms complex expressions into integrable forms, making the integration process more straightforward.
추천 영상:
7:17
Verifying Trig Equations as Identities

Verification by Differentiation

After finding an antiderivative, differentiating it should return the original integrand. This verification step ensures the correctness of the solution and helps identify any errors in the integration process.
추천 영상:
05:53
Finding Differentials
관련 실천
교과서 질문

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.


∫x⁻¹ᐟ³ dx

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교과서 질문

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.


∫(8y − 2 / y¹ᐟ⁴) dy

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교과서 질문

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.


∫(1 + tan²θ)dθ (Hint:1 + tan²θ = sec²θ)

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교과서 질문

Each of Exercises 67–88 gives the first derivative of a continuous function y=f(x). Find y'' and then use Steps 2–4 of the graphing procedure described in this section to sketch the general shape of the graph of f.

80. y' = 1 - cot²θ, for 0 < θ < π

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교과서 질문

Each of Exercises 67–88 gives the first derivative of a continuous function y=f(x). Find y'' and then use Steps 2–4 of the graphing procedure described in this section to sketch the general shape of the graph of f.

77. y' = cot(θ/2), for 0 < θ < 2π

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교과서 질문

Initial Value Problems


Solve the initial value problems in Exercises 71–90.


dy/dx = 1/x² + x, x > 0; y(2) = 1

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