Textbook Question
The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.
∫₀² (s + 1) / √(4 − s²) ds
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Ch. 8 - Techniques of Integration
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 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
Chapter 8, Problem 8.1.30
The integrals in Exercises 1–44 are in no particular order. Evaluate each integral using any algebraic method, trigonometric identity, or substitution you think is appropriate.
∫ 3 sinh(x/2 + ln 5) dx
Verified step by step guidance1
Recognize that the integral involves the hyperbolic sine function \( \sinh \left( \frac{x}{2} + \ln 5 \right) \). Recall that \( \sinh(u) = \frac{e^{u} - e^{-u}}{2} \), which can be useful for integration.
Let \( u = \frac{x}{2} + \ln 5 \). Then, compute \( du = \frac{1}{2} dx \), which implies \( dx = 2 du \). This substitution will simplify the integral.
Rewrite the integral in terms of \( u \): \( \int 3 \sinh(u) dx = \int 3 \sinh(u) \cdot 2 du = \int 6 \sinh(u) du \).
Integrate \( \sinh(u) \) with respect to \( u \). Recall that \( \int \sinh(u) du = \cosh(u) + C \). So, \( \int 6 \sinh(u) du = 6 \cosh(u) + C \).
Substitute back \( u = \frac{x}{2} + \ln 5 \) to express the answer in terms of \( x \): the integral is \( 6 \cosh \left( \frac{x}{2} + \ln 5 \right) + C \).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Hyperbolic Functions
Hyperbolic functions, such as sinh(x), are analogs of trigonometric functions but based on hyperbolas. The sinh function is defined as (e^x - e^(-x))/2 and has properties useful for integration, including linearity and known derivatives.
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Asymptotes of Hyperbolas
Integration by Substitution
Integration by substitution involves changing variables to simplify an integral. By letting u equal an inner function (e.g., u = x/2 + ln 5), the integral can be rewritten in terms of u, making it easier to evaluate.
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04:27Substitution With an Extra Variable
Linearity of Integration
The linearity property states that constants can be factored out of integrals and sums of functions can be integrated term-by-term. This allows the constant 3 to be factored out, simplifying the integral before applying substitution.
Recommended video:
07:17Linearization
Related Practice
Textbook Question
For Exercises 49–52, complete the square before using an appropriate trigonometric substitution.
∫ 1 / √(x² - 2x + 5) dx
Textbook Question
The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.
∫₋∞² (2 dx) / (x² + 4)
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Textbook Question
The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.
∫₋₈¹ dx / x^(1/3)
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Textbook Question
Evaluate the integrals in Exercises 31–56. Some integrals do not require integration by parts.
∫ (cos(√x))/(√x) dx
Textbook Question
Evaluate the integrals in Exercises 1–22.
∫ sin⁴(2x) cos(2x) dx