Textbook Question
Area functions for linear functions Consider the following functions Ζ and real numbers a (see figure).
b) Verify that A'(π) = Ζ(π).
Ζ(t) = 4t + 2 , a = 0
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Ch. 5 - Integration
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 5, Problem 5.5.15b
Use Table 5.6 to evaluate the following indefinite integrals.
(b) β« sec 5π tan 5π dπ
Verified step by step guidance1
Step 1: Recognize the integral β« sec(5π) tan(5π) dπ as a standard form listed in Table 5.6. The integral of sec(u) tan(u) with respect to u is sec(u).
Step 2: Identify the inner function u = 5π. This substitution is necessary because the argument of sec and tan is 5π, not just π.
Step 3: Compute the derivative of u with respect to π, which is du/dπ = 5. Rearrange to express dπ in terms of du: dπ = du/5.
Step 4: Substitute u = 5π and dπ = du/5 into the integral. The integral becomes (1/5) β« sec(u) tan(u) du.
Step 5: Apply the standard result from Table 5.6: β« sec(u) tan(u) du = sec(u). Replace u with 5π to return to the original variable, yielding (1/5) sec(5π) + 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 Integrals
Indefinite integrals represent a family of functions whose derivative is the integrand. They are expressed with a constant of integration, typically denoted as 'C'. The process of finding an indefinite integral is known as integration, which is the reverse operation of differentiation. Understanding the properties and rules of integration is essential for evaluating integrals accurately.
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Introduction to Indefinite Integrals
Integration Techniques
Various techniques exist for evaluating integrals, including substitution, integration by parts, and using trigonometric identities. For integrals involving trigonometric functions, recognizing patterns and applying specific formulas can simplify the process. Mastery of these techniques allows for the effective evaluation of more complex integrals, such as those involving secant and tangent functions.
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Trigonometric Integrals
Trigonometric integrals involve the integration of functions that include trigonometric identities, such as sine, cosine, secant, and tangent. Specific integrals, like β« sec(x) tan(x) dx, have known results that can be directly applied. Familiarity with these standard integrals, often found in integral tables, is crucial for quickly solving problems involving trigonometric functions.
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Related Practice
Textbook Question
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
(a) If Ζ is symmetric about the line π = 2 , then β«ββ΄ Ζ(π) dπ = 2 β«βΒ² Ζ(π) dπ.
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Textbook Question
Displacement from a velocity graph Consider the velocity function for an object moving along a line (see figure).
(b) Use geometry to find the displacement of the object between t = 0 and t = 2.
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Textbook Question
Free fall On October 14, 2012, Felix Baumgartner stepped off a balloon capsule at an altitude of almost 39 km above Earthβs surface and began his free fall. His velocity in m/s during the fall is given in the figure. It is claimed that Felix reached the speed of sound 34 seconds into his fall and that he continued to fall at supersonic speed for 30 seconds. (Source: http://www.redbullstratos.com)
(a) Divide the interval [34, 64] into n = 5 subintervals with the gridpoints xβ = 34 , xβ = 40 , xβ = 46 , xβ = 52 , xβ = 58 , and xβ = 64. Use left and right Riemann sums to estimate how far Felix fell while traveling at supersonic speed.
Textbook Question
Matching functions with area functions Match the functions Ζ, whose graphs are given in aβ d, with the area functions A (π) = β«βΛ£ Ζ(t) dt, whose graphs are given in AβD.
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Textbook Question
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
(b) Suppose Ζ is a negative increasing function, for π > 0 . Then the area function A(π) = β«βΛ£ Ζ(t) dt is a decreasing function of π .