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Theory and Applications
L’Hôpital’s Rule does not help with the limits in Exercises 69–76.
Try it—you just keep on cycling. Find the limits some other way.
73. lim (x → ∞) (2^x - 3^x) / (3^x + 4^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 7.3.41
Evaluate the integrals in Exercises 33–54.
∫ (e^(√r) / √r) dr
Guida verificata passo dopo passo1
Identify the integral to solve: \(\int \frac{e^{\sqrt{r}}}{\sqrt{r}} \, dr\).
Use substitution to simplify the integral. Let \(u = \sqrt{r}\), which means \(u = r^{1/2}\).
Differentiate \(u\) with respect to \(r\) to find \(du\): \(du = \frac{1}{2\sqrt{r}} \, dr\), or equivalently, \(dr = 2\sqrt{r} \, du\).
Rewrite the integral in terms of \(u\) by substituting \(\sqrt{r} = u\) and \(dr = 2u \, du\). The integral becomes \(\int \frac{e^{u}}{u} \cdot 2u \, du\).
Simplify the integral to \(\int 2 e^{u} \, du\), which is easier to integrate with respect to \(u\).
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Integration by Substitution
Integration by substitution is a method used to simplify integrals by changing variables. It involves identifying a part of the integrand as a new variable, which transforms the integral into a simpler form. This technique is especially useful when the integral contains a composite function, such as e^(√r).
Video consigliato:
04:27
Substitution With an Extra Variable
Exponential Functions
Exponential functions have the form e^x, where e is Euler's number. Their derivatives and integrals often involve the same exponential function multiplied by the derivative of the exponent. Understanding how to handle exponential functions with variable exponents is crucial for solving integrals like ∫ (e^(√r) / √r) dr.
Video consigliato:
6:13Exponential Functions
Chain Rule in Reverse (Differentiation and Integration)
The chain rule relates the derivative of a composite function to the derivatives of its inner and outer functions. In integration, applying the chain rule in reverse helps identify substitutions and integrals involving composite functions. Recognizing the derivative of √r within the integral guides the substitution process.
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05:02Intro to the Chain Rule
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In Exercises 7–26, find the derivative of y with respect to x, t, or θ, as appropriate.
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