Domanda del libro di testo
The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.
∫₋∞^∞ (x dx) / (x² + 4)^(3/2)
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Ch. 8 - Techniques of Integration
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 8, Problema 8.5.40
Evaluate the integrals in Exercises 39–54.
∫ (e⁴t + 2e²t - e^t) / (e²t + 1) dt
Guida verificata passo dopo passo1
Start by examining the integral \( \int \frac{e^{4t} + 2e^{2t} - e^{t}}{e^{2t} + 1} \, dt \). Notice that the numerator and denominator involve exponential functions with powers related to \( t \).
Make a substitution to simplify the expression. Let \( u = e^{2t} \). Then, compute \( du \) in terms of \( dt \): \( du = 2e^{2t} dt = 2u dt \), so \( dt = \frac{du}{2u} \).
Rewrite the integral in terms of \( u \). Express each term in the numerator and denominator using \( u \): \( e^{4t} = (e^{2t})^2 = u^2 \), \( 2e^{2t} = 2u \), and \( e^{t} = e^{t} \) (which can be expressed as \( e^{t} = (e^{2t})^{1/2} = u^{1/2} \)). Substitute these into the integral and replace \( dt \) with \( \frac{du}{2u} \).
Simplify the resulting integral in terms of \( u \), combining like terms and simplifying the fraction if possible. This should reduce the integral to a rational function of \( u \) and possibly \( u^{1/2} \).
Once simplified, split the integral into simpler parts if needed and integrate each term with respect to \( u \). After integrating, substitute back \( u = e^{2t} \) to express the answer in terms of \( t \).
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Integration of Rational Functions
This involves integrating expressions where one function is divided by another, often requiring algebraic manipulation such as polynomial division or substitution to simplify the integrand before integration.
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Intro to Rational Functions
Exponential Functions and Their Properties
Understanding the rules of exponents, such as e^{a} * e^{b} = e^{a+b}, and how to rewrite expressions with exponential terms is crucial for simplifying the integrand and identifying substitution opportunities.
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Percorso guidato
06:21Properties of Functions
Substitution Method in Integration
This technique involves changing variables to simplify the integral, often by letting a part of the integrand equal a new variable, which transforms the integral into a more manageable form.
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07:33Euler's Method
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