Textbook Question
Solve the differential equation in Exercises 9–22.
13. (dy/dx) = √y cos²√y
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 7.7.45
Evaluate the integrals in Exercises 41–60.
45. ∫tanh(x/7)dx
Verified step by step guidance1
Recall that the integral of the hyperbolic tangent function \( \tanh(u) \) with respect to \( u \) is \( \int \tanh(u) \, du = \ln|\cosh(u)| + C \).
Identify the inner function inside the hyperbolic tangent: here, \( u = \frac{x}{7} \).
Use substitution: let \( u = \frac{x}{7} \), then \( du = \frac{1}{7} dx \) or equivalently \( dx = 7 \, du \).
Rewrite the integral in terms of \( u \): \( \int \tanh\left(\frac{x}{7}\right) dx = \int \tanh(u) \cdot 7 \, du = 7 \int \tanh(u) \, du \).
Integrate using the known formula: \( 7 \int \tanh(u) \, du = 7 \ln|\cosh(u)| + C \), then substitute back \( u = \frac{x}{7} \) to get the final expression.
Verified video answer for a similar problem:
This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
1mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Hyperbolic Tangent Function (tanh)
The hyperbolic tangent function, tanh(x), is defined as (e^x - e^(-x)) / (e^x + e^(-x)). It is an odd function with values ranging between -1 and 1. Understanding its properties helps in recognizing integrals involving tanh and applying appropriate integration techniques.
Recommended video:
Guided course
05:13
Slopes of Tangent Lines
Integration of Hyperbolic Functions
Integrating hyperbolic functions often involves using their definitions or known antiderivatives. For tanh(x), the integral is ln(cosh(x)) + C. Recognizing these standard integrals simplifies the process and avoids complex substitutions.
Recommended video:
Guided course
5:50Asymptotes of Hyperbolas
Substitution Method in Integration
When the integrand contains a function of a linear expression like tanh(x/7), substitution helps simplify the integral. Setting u = x/7 transforms the integral into a standard form, making it easier to integrate and then revert to the original variable.
Recommended video:
07:33Euler's Method
Related Practice
Textbook Question
In Exercises 27–32, find dy/dx.
ln y = e^y sinx
Textbook Question
Which of the functions graphed in Exercises 1–6 are one-to-one, and which are not?
Textbook Question
Evaluate the integrals in Exercises 41–60.
57. ∫(from 1 to 2)cosh(ln t)/t dt
1
views
Textbook Question
37. Plutonium-239 The half-life of the plutonium isotope is 24,360 years. If 10 g of plutonium is released into the atmosphere by a nuclear accident, how many years will it take for 80% of the isotope to decay?
Textbook Question
Indeterminate Powers and Products
Find the limits in Exercises 53–68.
63. lim (x → ∞) ((x + 2)/(x - 1))^x