Textbook Question
In Exercises 1–24, find the derivative of y with respect to the appropriate variable.
19. y = t arctan(t) - 1/2 ln(t)
Ch. 7 - Transcendental Functions
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 Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 7, Problem 7.PE.58
Evaluate the integrals in Exercises 31–78.
58. ∫(from 0 to ln9)e^θ(e^θ-1)^(1/2) dθ
Verified step by step guidance1
Start by examining the integral: \(\int_0^{\ln 9} e^{\theta} \left(e^{\theta} - 1\right)^{1/2} \, d\theta\). Notice that the integrand contains \(e^{\theta}\) and a function of \(e^{\theta} - 1\), which suggests a substitution involving \(e^{\theta}\).
Let’s perform the substitution \(u = e^{\theta} - 1\). Then, differentiate both sides with respect to \(\theta\) to find \(du\): \(du = e^{\theta} d\theta\).
Rewrite the integral in terms of \(u\). Since \(du = e^{\theta} d\theta\), the integral becomes \(\int \sqrt{u} \, du\) after changing the limits accordingly.
Change the limits of integration from \(\theta\) to \(u\): when \(\theta = 0\), \(u = e^0 - 1 = 0\); when \(\theta = \ln 9\), \(u = e^{\ln 9} - 1 = 9 - 1 = 8\).
Now, the integral is \(\int_0^8 u^{1/2} \, du\). This is a standard power integral, which can be integrated using the formula \(\int u^n du = \frac{u^{n+1}}{n+1} + C\) for \(n \neq -1\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Definite Integrals
A definite integral calculates the net area under a curve between two specified limits. It involves evaluating the antiderivative at the upper and lower bounds and subtracting these values. This process yields a numerical value representing the accumulated quantity over the interval.
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Definition of the Definite Integral
Substitution Method
The substitution method simplifies integrals by changing variables to transform the integral into a more manageable form. By letting a new variable equal a function inside the integral, the integral often becomes easier to evaluate. This technique is especially useful when the integral contains composite functions.
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07:33Euler's Method
Exponential Functions and Their Properties
Exponential functions, like e^θ, have unique properties such as their derivative and integral being proportional to themselves. Understanding how to manipulate expressions involving exponentials, including their growth and composition with other functions, is essential for solving integrals involving these terms.
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Related Practice
Textbook Question
Evaluate the integrals in Exercises 31–78.
43. ∫tan(ln v)/v dv
Textbook Question
In Exercises 1–24, find the derivative of y with respect to the appropriate variable.
23. y = arccsc(secθ), 0<θ<π/2
Textbook Question
In Exercises 1–24, find the derivative of y with respect to the appropriate variable.
15. y = sin⁻¹√(1-u²), 0<u<1
Textbook Question
In Exercises 129–132 solve the initial value problem.
129. dy/dx = e^(-x-y-2), y(0) = -2
Textbook Question
In Exercises 115 and 116, find the absolute maximum and minimum values of each function on the given interval.
116. y = 10x (2 - ln(x)), (0, e²]"133. Find the absolute maximum value of
f(x) = x^2 * ln(1/x)
and say where it is assumed