Textbook Question
In Exercises 59–86, find the derivative of y with respect to the given independent variable.
73. y = log₄ x + log₄ x²
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 7.3.87
Evaluate the integrals in Exercises 87–96.
87. ∫ 5ˣ dx
Verified step by step guidance1
Recognize that the integral involves an exponential function with base 5, which can be written as \$5^x$.
Recall the general formula for integrating an exponential function with base \(a\): \(\int a^x \, dx = \frac{a^x}{\ln(a)} + C\), where \(a > 0\) and \(a \neq 1\).
Apply this formula to the integral \(\int 5^x \, dx\), substituting \(a = 5\).
Write the integral as \(\int 5^x \, dx = \frac{5^x}{\ln(5)} + C\), where \(C\) is the constant of integration.
This expression represents the antiderivative of \$5^x$.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Exponential Functions
An exponential function has the form f(x) = a^x, where the base a is a positive constant. Understanding how these functions behave and their properties is essential for integrating expressions like 5^x.
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Exponential Functions
Integration of Exponential Functions
The integral of an exponential function a^x with respect to x is (a^x) / ln(a) + C, where ln(a) is the natural logarithm of the base. This formula is crucial for solving integrals involving exponential terms.
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05:11Integrals of General Exponential Functions
Natural Logarithm (ln)
The natural logarithm ln(x) is the inverse of the exponential function e^x. It appears in the integration formula of exponential functions and is important for simplifying and understanding the results of such integrals.
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05:18Derivative of the Natural Logarithmic Function
Related Practice
Textbook Question
Evaluate the integrals in Exercises 39–56.
56. ∫sec(x)dx/√(ln(sec(x)+tan(x)))
Textbook Question
Evaluate the integrals in Exercises 87–96.
95. ∫₂⁴ x^(2x) (1 + ln x) dx
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Textbook Question
In Exercises 7–38, find the derivative of y with respect to x, t, or θ, as appropriate.
37. ∫(from x²/2 to x²)ln(√t)dt
Textbook Question
80. Volume The region enclosed by the curve y=sech(x), the x-axis, and the lines x=±ln√3 is revolved about the x-axis to generate a solid. Find the volume of the solid.
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Textbook Question
In Exercises 13–24, find the derivative of y with respect to the appropriate variable.
15. y = 2√t tanh(√t)