Ch. 7 - Transcendental Functions

Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook

Hass 15th Edition

Ch. 7 - Transcendental Functions

Problem 7.3.37

Chapter 7, Problem 7.3.37

Evaluate the integrals in Exercises 33–54.
∫8e^(x+1) dx

Verified step by step guidance

Recognize that the integral is of the form $\int 8 e^{x+1} \, dx$, where the integrand is an exponential function with a linear argument in the exponent.

Use the property of exponents to rewrite the integrand as \$8 e^{x} e^{1}$, or equivalently factor out the constant $e^{1}$ since it does not depend on $x$.

Rewrite the integral as $8 e^{1} \int e^{x} \, dx$, separating the constant multiplier from the integral.

Recall the integral formula for the exponential function: $\int e^{x} \, dx = e^{x} + C$, where $C$ is the constant of integration.

Apply the integral formula and multiply back by the constants to express the antiderivative as $8 e^{1} e^{x} + C$, which can be combined as $8 e^{x+1} + C$.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Integration of Exponential Functions

Integrating exponential functions involves reversing differentiation. For functions like e^(ax+b), the integral is (1/a)e^(ax+b) + C, where a and b are constants. Recognizing the inner function helps apply the correct formula.

Constant Multiple Rule

The constant multiple rule states that a constant factor can be pulled out of the integral. For example, ∫k*f(x) dx = k*∫f(x) dx. This simplifies integration by isolating constants from variable expressions.

Indefinite Integrals and Integration Constants

Indefinite integrals represent families of functions differing by a constant, denoted as + C. This constant accounts for all possible antiderivatives since differentiation of a constant is zero.

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Introduction to Indefinite Integrals

Related Practice

Textbook Question

In Exercises 25–36, find the derivative of y with respect to the appropriate variable.

27. y = (1 - θ)tanh⁻¹(θ)

Textbook Question

Since the hyperbolic functions can be expressed in terms of exponential functions, it is possible to express the inverse hyperbolic functions in terms of logarithms, as shown in the following table.

sinh⁻¹x = ln(x + √(x² + 1)), -∞ < x < ∞

cosh⁻¹x = ln(x + √(x² - 1)), x ≥ 1

tanh⁻¹x = (1/2)ln((1+x)/(1-x)), |x| < 1

sech⁻¹x = ln((1+√(1-x²))/x), 0 < x ≤ 1

csch⁻¹x = ln(1/x + √(1+x²)/|x|), x ≠ 1

coth⁻¹x = (1/2)ln((x+1)/(x-1)), |x| > 1

Use these formulas to express the numbers in Exercises 61–66 in terms of natural logarithms.

63. tanh⁻¹(-1/2)

Textbook Question

Evaluate the integrals in Exercises 97–110.

109. ∫ (dx / (x log₁₀x))

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