Textbook Question
In Exercises 1–4, solve for t.
e^(sqrt(t)) = x^2
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 7.3.113
Evaluate the integrals in Exercises 111–114.
113. ∫₁^(1/x) (1 / t) dt,x > 0
Verified step by step guidance1
Identify the integral to be evaluated: \(\int_{1}^{\frac{1}{x}} \frac{1}{t} \, dt\) where \(x > 0\).
Recall the antiderivative of the integrand \(\frac{1}{t}\), which is \(\ln|t|\).
Apply the Fundamental Theorem of Calculus, which states that \(\int_{a}^{b} f(t) \, dt = F(b) - F(a)\), where \(F\) is an antiderivative of \(f\).
Substitute the limits of integration into the antiderivative: \(\ln\left| \frac{1}{x} \right| - \ln|1|\).
Simplify the expression using logarithm properties, such as \(\ln\left( \frac{1}{x} \right) = -\ln(x)\) and \(\ln(1) = 0\), to express the integral in terms of \(x\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Definite Integral and Variable Limits
A definite integral calculates the net area under a curve between two limits. When the limits themselves depend on a variable, the integral becomes a function of that variable, requiring careful evaluation of the integral with respect to those limits.
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Definition of the Definite Integral
Integral of 1/t
The integral of 1/t with respect to t is the natural logarithm function, ln|t|, plus a constant. This fundamental integral is essential for solving problems involving logarithmic functions and appears frequently in calculus.
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Properties of the Natural Logarithm
The natural logarithm has properties such as ln(a/b) = ln(a) - ln(b), which simplify expressions involving logarithms. Understanding these properties helps in manipulating and simplifying the results of integrals with variable limits.
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Related Practice
Textbook Question
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Textbook Question
Solve the differential equation in Exercises 9–22.
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