Textbook Question
Checking Antiderivative Formulas
Verify the formulas in Exercises 57–62 by differentiation.
∫(3x + 5)⁻² dx = −(3x + 5)⁻¹/3 + C
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Ch. 4 - Applications of Derivatives
Hass15th EditionThomas' CalculusISBN: 9780137616077
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Table of contents
- 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 Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 4, Problem 4.7.47
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 + cos 4t)/2 dt
Verified step by step guidance1
Start by rewriting the integral to separate the terms inside the integral: \(\int \frac{1 + \cos 4t}{2} \, dt = \int \frac{1}{2} \, dt + \int \frac{\cos 4t}{2} \, dt\).
Integrate the first term \(\int \frac{1}{2} \, dt\) which is a constant multiple of \(dt\). Recall that \(\int a \, dt = at\) for constant \(a\).
For the second term \(\int \frac{\cos 4t}{2} \, dt\), factor out the constant \(\frac{1}{2}\) to get \(\frac{1}{2} \int \cos 4t \, dt\).
Use the substitution rule or recall the integral formula \(\int \cos (kt) \, dt = \frac{1}{k} \sin (kt) + C\). Apply this with \(k=4\) to find the antiderivative of \(\cos 4t\).
Combine the results from both integrals and add the constant of integration \(C\) to write the most general antiderivative.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Indefinite Integral and Antiderivative
An indefinite integral represents the most general antiderivative of a function, including a constant of integration. It reverses differentiation, providing a family of functions whose derivative equals the integrand.
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Integration of Trigonometric Functions
Integrating trigonometric functions like cosine involves using known integral formulas, such as ∫cos(ax) dx = (1/a) sin(ax) + C. Recognizing and applying these formulas simplifies finding antiderivatives.
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Verification by Differentiation
After finding an antiderivative, differentiating it should return the original integrand. This step confirms the correctness of the integral and helps adjust any initial guesses.
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