Textbook Question
Using properties of integrals Use the value of the first integral I to evaluate the two given integrals.
I = β«βΒΉ (πΒ³ β 2π) dπ = β3/4
(a) β«βΒΉ (4πβ2πΒ³) dπ
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 5.5.15a
Use Table 5.6 to evaluate the following indefinite integrals.
(a) β« eΒΉβ°Λ£ dπ
Verified step by step guidance1
Step 1: Recognize that the integral involves the exponential function e raised to a linear term (10x). The formula for the integral of e^(kx) is (1/k) * e^(kx) + C, where k is a constant and C is the constant of integration.
Step 2: Identify the constant k in the exponent. In this case, k = 10 because the exponent is 10x.
Step 3: Apply the formula for the integral of e^(kx). Substitute k = 10 into the formula, resulting in (1/10) * e^(10x) + C.
Step 4: Write the result in terms of the indefinite integral. The integral β« e^(10x) dx simplifies to (1/10) * e^(10x) + C.
Step 5: Remember to include the constant of integration (C) in your final answer, as this is an indefinite integral.
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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 the integral sign followed by the function and the differential, and they include a constant of integration (C) since the derivative of a constant is zero. Understanding how to evaluate indefinite integrals is crucial for solving problems in calculus, as they provide the antiderivative of a function.
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Exponential Functions
Exponential functions are mathematical functions of the form f(x) = a^x, where 'a' is a constant and 'x' is the variable. In calculus, the natural exponential function e^x is particularly important due to its unique property that the derivative of e^x is e^x itself. This property simplifies the process of integration, especially when dealing with integrals involving exponential terms.
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Integration Techniques
Integration techniques are methods used to evaluate integrals that may not be straightforward. Common techniques include substitution, integration by parts, and using integral tables. Familiarity with these techniques allows students to tackle a variety of integrals, including those involving exponential functions, and is essential for effectively solving calculus problems.
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Related Practice
Textbook Question
Bounds on an integral Suppose Ζ is continuous on [a, b] with Ζ''(π) > 0 on the interval. It can be shown that (bβa) Ζ [(a + b) /2] β€ β«βα΅ Ζ(π) dπ β€ (bβa) [ (Ζ(a) + Ζ(b)) /2]
(a) Assuming Ζ is nonnegative on [a, b], draw a figure to illustrate the geometric meaning of these inequalities. Discuss your conclusions. b.
Textbook Question
The velocity in ft/s of an object moving along a line is given by v = Ζ(t) on the interval 0 β€ t β€ 8 (see figure), where t is measured in seconds.
a) Divide the interval [0,8] into n = 2 subintervals, [0,4] and [4,8]. On each subinterval, assume the object moves at a constant velocity equal to the value of v evaluated at the midpoint of the subinterval, and use these approximations to estimate the displacement of the object on [0,8] (see part (a) of the figure)
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Textbook Question
Working with area functions Consider the function Ζ and the points a, b, and c.
(a) Find the area function A (π) = β«βΛ£ Ζ(t) dt using the Fundamental Theorem.
Ζ(π) = β 12π (πβ1) (πβ 2) ; a = 0 , b = 1 , c = 2
Textbook Question
Area functions The graph of Ζ is shown in the figure. Let A(x) = β«ββΛ£ Ζ(t) dt and F(x) = β«βΛ£ Ζ(t) dt be two area functions for Ζ. Evaluate the following area functions.
(a) A (β2)
Textbook Question
{Use of Tech} Midpoint Riemann sums with a calculator Consider the following definite integrals.
(a) Write the midpoint Riemann sum in sigma notation for an arbitrary value of n.
β«ββ΄ 2βπ dπ