Textbook Question
Evaluate the integrals in Exercises 33–54.
∫ 2t e^(-t²) dt
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 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 7, Problem 7.3.112
Evaluate the integrals in Exercises 111–114.
112. ∫₁^(eˣ) (1 / t) dt
Verified step by step guidance1
Recognize that the integral is a definite integral with variable upper limit: \(\int_1^{e^x} \frac{1}{t} \, dt\).
Recall the Fundamental Theorem of Calculus Part 1, which states that if \(F(x) = \int_a^{g(x)} f(t) \, dt\), then \(F'(x) = f(g(x)) \cdot g'(x)\).
Identify the integrand function as \(f(t) = \frac{1}{t}\) and the upper limit as \(g(x) = e^x\).
Apply the Fundamental Theorem of Calculus to differentiate the integral with respect to \(x\): \(\frac{d}{dx} \int_1^{e^x} \frac{1}{t} \, dt = \frac{1}{e^x} \cdot \frac{d}{dx} (e^x)\).
Calculate the derivative of the upper limit: \(\frac{d}{dx} (e^x) = e^x\), then multiply by the integrand evaluated at \(e^x\) to get the derivative of the integral.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Definite Integral with Variable Limits
A definite integral with variable limits involves integrating a function where the upper or lower limit is a function of a variable, such as e^x. Understanding how to handle these limits is essential for evaluating the integral correctly.
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Definition of the Definite Integral
Fundamental Theorem of Calculus
This theorem connects differentiation and integration, stating that if F is an antiderivative of f, then the definite integral of f from a to b equals F(b) - F(a). It allows evaluating integrals by finding antiderivatives.
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06:11Fundamental Theorem of Calculus Part 1
Integral of 1/t
The integral of 1/t with respect to t is the natural logarithm function, ln|t| + C. Recognizing this integral form is crucial for solving the given problem, especially when evaluating definite integrals involving 1/t.
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09:15Tabular Integration by Parts Example 6
Related Practice
Textbook Question
In Exercises 59–86, find the derivative of y with respect to the given independent variable.
85. y = log₂(8t^(ln 2))
Textbook Question
Use l’Hôpital’s rule to find the limits in Exercises 7–52.
32. lim (x → 0) (3^x - 1) / (2^x - 1)
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Textbook Question
Use l’Hôpital’s rule to find the limits in Exercises 7–52.
37. lim (y → 0) (√(5y + 25) - 5) / y
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Textbook Question
Theory and Applications
L’Hôpital’s Rule does not help with the limits in Exercises 69–76.
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71. lim (x → (π/2)⁻) sec x / tan x
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Textbook Question
In Exercises 115–126, use logarithmic differentiation or the method in Example 6 to find the derivative of y with respect to the given independent variable.
117. y = (√t)ᵗ