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Ch. 7 - Transcendental Functions
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 7 - Transcendental FunctionsProblem 7.PE.35
Chapter 7, Problem 7.PE.35

Evaluate the integrals in Exercises 31–78.
35. ∫sec²x e^(tan x)dx

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1
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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Slopes 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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Integrals of General Exponential Functions
Related Practice
Textbook Question

Evaluate the integrals in Exercises 31–78.

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In Exercises 1–24, find the derivative of y with respect to the appropriate variable.

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Evaluate the integrals in Exercises 31–78.

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Textbook Question

In Exercises 79–84, solve for y.

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