Textbook Question
Evaluate the integrals in Exercises 31–78.
31. ∫e^x sin(e^x)dx
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.PE.35
Evaluate the integrals in Exercises 31–78.
35. ∫sec²x e^(tan x)dx
Verified step by step guidance1
Recognize that the integral involves the function \(e^{\tan x}\) multiplied by \(\sec^2 x\). This suggests a substitution related to the derivative of \(\tan x\).
Recall that the derivative of \(\tan x\) is \(\sec^2 x\), which matches the factor multiplying \(e^{\tan x}\) in the integral.
Set the substitution \(u = \tan x\), so that \(du = \sec^2 x \, dx\). This allows us to rewrite the integral in terms of \(u\).
Rewrite the integral as \(\int e^u \, du\), which is a standard integral involving the exponential function.
Integrate \(e^u\) with respect to \(u\) to get \(e^u + C\), then substitute back \(u = \tan x\) to express the answer 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.
Integration by Substitution
Integration by substitution is a method used to simplify integrals by changing variables. It involves identifying a part of the integrand whose derivative is also present, allowing the integral to be rewritten in terms of a new variable. This technique is especially useful when the integral contains a composite function.
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Substitution With an Extra Variable
Derivative of the Tangent Function
The derivative of tan(x) is sec²(x). Recognizing this relationship helps in integration problems where sec²(x) appears alongside functions of tan(x). This connection often suggests using tan(x) as a substitution variable to simplify the integral.
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05:13Slopes of Tangent Lines
Exponential Functions in Integration
Exponential functions like e^(tan x) maintain their form under differentiation and integration, making them straightforward to handle once the substitution is made. Understanding how to integrate expressions involving exponentials combined with trigonometric functions is key to solving such integrals.
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05:11Integrals of General Exponential Functions
Related Practice
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