Domanda del libro di testo
Evaluate the integrals in Exercises 33–54.
∫(from ln4 to ln9)e^(x/2)dx
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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 7.2.41
Evaluate the integrals in Exercises 39–56.
41. ∫2y dy/(y²-25)
Guida verificata passo dopo passo1
Identify the integral to solve: \(\int \frac{2y}{y^{2} - 25} \, dy\).
Recognize that the denominator is a quadratic expression that can be factored as \(y^{2} - 25 = (y - 5)(y + 5)\), but before factoring, check if substitution is more straightforward.
Use substitution by letting \(u = y^{2} - 25\), then compute \(du = 2y \, dy\), which matches the numerator times \(dy\) exactly.
Rewrite the integral in terms of \(u\): \(\int \frac{2y}{y^{2} - 25} \, dy = \int \frac{1}{u} \, du\).
Integrate \(\int \frac{1}{u} \, du\) to get \(\ln|u| + C\), then substitute back \(u = y^{2} - 25\) to express the answer in terms of \(y\).
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Integration of Rational Functions
This involves integrating functions expressed as the ratio of two polynomials. Techniques often include algebraic manipulation such as factoring or partial fraction decomposition to simplify the integral into manageable parts.
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Intro to Rational Functions
Partial Fraction Decomposition
A method used to break down complex rational expressions into simpler fractions that are easier to integrate. It is especially useful when the denominator can be factored into linear or quadratic terms.
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Percorso guidato
10:07Partial Fraction Decomposition: Distinct Linear Factors
Substitution Method
A technique where a substitution is made to simplify the integral, often by letting a part of the integrand equal a new variable. This can transform the integral into a basic form that is straightforward to evaluate.
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07:33Euler's Method
Pratica correlata
Domanda del libro di testo
41. Cooling soup Suppose that a cup of soup cooled from 90°C to 60°C after 10 min in a room where the temperature was 20°C. Use Newton’s Law of Cooling to answer the following questions.
a. How much longer would it take the soup to cool to 35°C?
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Domanda del libro di testo
In Exercises 59–86, find the derivative of y with respect to the given independent variable.
85. y = log₂(8t^(ln 2))
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Domanda del libro di testo
In Exercises 7–26, find the derivative of y with respect to x, t, or θ, as appropriate.
y = ln(e^(θ)/(1+e^θ))
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Domanda del libro di testo
Evaluate the integrals in Exercises 39–56.
52. ∫(from π/4 to π/2)cot(t)dt
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Domanda del libro di testo
In Exercises 7–26, find the derivative of y with respect to x, t, or θ, as appropriate.
y = e^(θ)(sinθ + cosθ)
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