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Ch. 4 - Applications of the Derivative
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 4 - Applications of the DerivativeProblem 4.R.95
Chapter 4, Problem 4.R.95

90–103. Indefinite integrals Determine the following indefinite integrals.


∫(1 + 3 cosΘ) dΘ

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Step 1: Recognize that the integral is in the form ∫(1 + 3 cosΘ) dΘ, which can be split into two separate integrals: ∫1 dΘ + ∫3 cosΘ dΘ.
Step 2: Solve the first integral ∫1 dΘ. The integral of 1 with respect to Θ is simply Θ, because the derivative of Θ is 1.
Step 3: Solve the second integral ∫3 cosΘ dΘ. Use the constant multiple rule, which allows you to factor out the constant 3, resulting in 3∫cosΘ dΘ.
Step 4: Recall the integral of cosΘ with respect to Θ is sinΘ. Therefore, ∫cosΘ dΘ = sinΘ, and 3∫cosΘ dΘ = 3sinΘ.
Step 5: Combine the results from Step 2 and Step 4. The indefinite integral ∫(1 + 3 cosΘ) dΘ is Θ + 3sinΘ + C, where C is the constant of integration.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Indefinite Integral

An indefinite integral represents a family of functions whose derivative is the integrand. It is expressed without limits of integration and includes a constant of integration, typically denoted as 'C'. The process of finding an indefinite integral is often referred to as antidifferentiation.
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Basic Integration Rules

Basic integration rules are fundamental formulas used to compute integrals. For example, the integral of a constant 'a' is 'aΘ', and the integral of cos(Θ) is sin(Θ). Understanding these rules is essential for solving integrals efficiently and accurately.
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Trigonometric Functions

Trigonometric functions, such as sine and cosine, are periodic functions that arise in various mathematical contexts, including calculus. Their properties, such as periodicity and symmetry, play a crucial role in integration, especially when dealing with integrals involving trigonometric expressions.
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Related Practice
Textbook Question

Folded boxes


a. Squares with sides of length x are cut out of each corner of a rectangular piece of cardboard measuring 5 ft by 8 ft. The resulting piece of cardboard is then folded into a box without a lid. Find the volume of the largest box that can be formed in this way.

Textbook Question

82–89. Comparing growth rates Determine which of the two functions grows faster, or state that they have comparable growth rates.


2ˣ and 4ˣ⸍²

Textbook Question

Sketch a graph of a function f with the following properties.


f' < 0 and f" < 0, for x < 3

Textbook Question

24–34. Curve sketching Use the guidelines given in Section 4.4 to make a complete graph of the following functions on their domains or on the given interval. Use a graphing utility to check your work.


{Use of Tech} ƒ(x) = x (x -1)e⁻ˣ

Textbook Question

Use the graphs of ƒ' and ƒ" to complete the following steps. <IMAGE>

b. Determine the locations of the inflection points of f and the intervals on which f is concave up or concave down.

Textbook Question

Suppose the objective function P= xy is subject to the constraint 10x + y = 100, where x and y are real numbers.


a. Eliminate the variable y from the objective function so that P is expressed as a function of one variable x.