Domanda del libro di testo
128. Derive the formula dy/dx = 1/(1+x²) for the derivative of y = arctan(x) by differentiating both sides of the equivalent equation tan(y)=x.
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Ch. 7 - Transcendental Functions
Hass15th EditionThomas' CalculusISBN: 9780137616077
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TOC.tableOfContents
- 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
Capitolo 7, Problema 7.6.73
Evaluate the integrals in Exercises 53–76.
73. ∫(from 0 to ln√3) e^x dx/(1+e^(2x))
Guida verificata passo dopo passo1
Recognize the integral to evaluate: \(\int_0^{\ln \sqrt{3}} \frac{e^x}{1 + e^{2x}} \, dx\).
Make a substitution to simplify the integral. Let \(u = e^x\). Then, the differential is \(du = e^x \, dx\), which means \(e^x \, dx = du\).
Rewrite the integral in terms of \(u\). The limits change as follows: when \(x = 0\), \(u = e^0 = 1\); when \(x = \ln \sqrt{3}\), \(u = e^{\ln \sqrt{3}} = \sqrt{3}\). The integral becomes \(\int_1^{\sqrt{3}} \frac{1}{1 + u^2} \, du\).
Recognize that \(\int \frac{1}{1 + u^2} \, du\) is the standard integral for \(\arctan u + C\). So, the integral evaluates to \(\arctan u\) evaluated from \(1\) to \(\sqrt{3}\).
Apply the Fundamental Theorem of Calculus by substituting the limits back into \(\arctan u\) to express the definite integral as \(\arctan(\sqrt{3}) - \arctan(1)\).
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Definite Integrals
A definite integral calculates the net area under a curve between two specific limits. It involves evaluating the antiderivative at the upper and lower bounds and subtracting these values. This concept is essential for solving integrals with given limits, such as from 0 to ln(√3).
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Percorso guidato
05:43
Definition of the Definite Integral
Substitution Method
The substitution method simplifies integrals by changing variables to transform the integral into a more manageable form. It often involves setting a part of the integrand as a new variable, which helps in integrating complex expressions like those involving exponential functions.
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07:33Euler's Method
Properties of Exponential Functions
Exponential functions, such as e^x, have unique properties including their derivatives and integrals being proportional to themselves. Understanding how to manipulate expressions like e^(2x) and their relationships is crucial for simplifying and evaluating integrals involving exponentials.
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06:21Properties of Functions
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