Textbook Question
Evaluate the integrals in Exercises 53–76.
61. ∫(from 0 to 2)dt/√(8+2t²)
1
views
Ch. 7 - Transcendental Functions
Hass15th EditionThomas' CalculusISBN: 9780137616077
Not the one you use?Change textbookSelect a different textbook
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 7, Problem 7.7.41
Evaluate the integrals in Exercises 41–60.
41. ∫sinh(2x)dx
Verified step by step guidance1
Recall the definition of the hyperbolic sine function: \(\sinh(x) = \frac{e^{x} - e^{-x}}{2}\).
Use the linearity of the integral to write \(\int \sinh(2x) \, dx\) as an integral involving exponentials: \(\int \sinh(2x) \, dx = \int \frac{e^{2x} - e^{-2x}}{2} \, dx\).
Split the integral into two separate integrals: \(\frac{1}{2} \int e^{2x} \, dx - \frac{1}{2} \int e^{-2x} \, dx\).
Integrate each exponential term using the formula \(\int e^{ax} \, dx = \frac{1}{a} e^{ax} + C\): so \(\int e^{2x} \, dx = \frac{1}{2} e^{2x} + C\) and \(\int e^{-2x} \, dx = -\frac{1}{2} e^{-2x} + C\).
Combine the results and simplify to express the integral back in terms of hyperbolic functions, then add the constant of integration \(C\).
Verified video answer for a similar problem:
This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
1mWas this helpful?
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) and cosh(x), are analogs of trigonometric functions but based on hyperbolas. The function sinh(x) is defined as (e^x - e^(-x))/2 and has properties similar to sine, useful in integration and differentiation.
Recommended video:
5:50
Asymptotes of Hyperbolas
Integration of Hyperbolic Functions
Integrating hyperbolic functions involves using their definitions or known integral formulas. For example, the integral of sinh(ax) with respect to x is (1/a)cosh(ax) + C, where a is a constant, leveraging the derivative relationship between sinh and cosh.
Recommended video:
5:50Asymptotes of Hyperbolas
Constant of Integration
When evaluating indefinite integrals, it is essential to include the constant of integration, denoted as C, because integration is the inverse of differentiation and the original function could differ by a constant.
Recommended video:
06:18Integration by Parts for Definite Integrals
Related Practice
Textbook Question
In Exercises 7–10, determine from its graph if the function is one-to-one.
f(x) = 1 - x/2, x ≤ 0
x/(x + 2), x > 0
Textbook Question
In Exercises 7–26, find the derivative of y with respect to x, t, or θ, as appropriate.
y = cos(e^(-θ^2))
Textbook Question
Evaluate the integrals in Exercises 33–54.
53. ∫ (e^r / (1 + e^r)) dr
Textbook Question
In Exercises 27–32, find dy/dx.
3+siny = y-x^3
Textbook Question
Evaluate the integrals in Exercises 97–110.
107. ∫₀⁹ (2 log₁₀(x + 1) / (x + 1)) dx
1
views