Textbook Question
Evaluate the integrals in Exercises 31–78.
31. ∫e^x sin(e^x)dx
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 Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 7, Problem 7.PE.93
Use l’Hôpital’s Rule to find the limits in Exercises 85–108.
93. lim(x→0) (csc(x) - cot(x))
Verified step by step guidance1
First, recognize that as \(x \to 0\), both \(\csc(x)\) and \(\cot(x)\) approach forms that may lead to an indeterminate expression. Specifically, \(\csc(x) = \frac{1}{\sin(x)}\) and \(\cot(x) = \frac{\cos(x)}{\sin(x)}\), so the expression \(\csc(x) - \cot(x)\) can be rewritten as \(\frac{1}{\sin(x)} - \frac{\cos(x)}{\sin(x)}\).
Combine the terms over a common denominator to simplify the expression: \(\csc(x) - \cot(x) = \frac{1 - \cos(x)}{\sin(x)}\).
Evaluate the limit \(\lim_{x \to 0} \frac{1 - \cos(x)}{\sin(x)}\). Direct substitution gives \(\frac{1 - 1}{0} = \frac{0}{0}\), which is an indeterminate form, so l’Hôpital’s Rule applies.
Apply l’Hôpital’s Rule by differentiating the numerator and denominator separately with respect to \(x\): differentiate numerator \(1 - \cos(x)\) to get \(\sin(x)\), and differentiate denominator \(\sin(x)\) to get \(\cos(x)\).
Rewrite the limit as \(\lim_{x \to 0} \frac{\sin(x)}{\cos(x)}\) and then evaluate this new limit by direct substitution.
Verified video answer for a similar problem:
This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
3mWas this helpful?
Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
l’Hôpital’s Rule
l’Hôpital’s Rule is a method for evaluating limits that result in indeterminate forms like 0/0 or ∞/∞. It states that the limit of a ratio of functions can be found by taking the limit of the ratio of their derivatives, provided certain conditions are met.
Recommended video:
Guided course
5:50
Power Rules
Trigonometric Functions and Their Limits
Understanding the behavior of trigonometric functions such as csc(x) and cot(x) near zero is essential. Knowing standard limits like sin(x) ≈ x near zero helps simplify expressions and identify indeterminate forms.
Recommended video:
6:04Introduction to Trigonometric Functions
Derivative of Trigonometric Functions
To apply l’Hôpital’s Rule, one must compute derivatives of csc(x) and cot(x). The derivatives are -csc(x)cot(x) and -csc²(x), respectively, which are crucial for simplifying the limit expression after differentiation.
Recommended video:
06:35Derivatives of Other Inverse Trigonometric Functions
Related Practice
Textbook Question
In Exercises 1–24, find the derivative of y with respect to the appropriate variable.
19. y = t arctan(t) - 1/2 ln(t)
Textbook Question
Evaluate the integrals in Exercises 31–78.
35. ∫sec²x e^(tan x)dx
Textbook Question
In Exercises 1–24, find the derivative of y with respect to the appropriate variable.
15. y = sin⁻¹√(1-u²), 0<u<1
Textbook Question
Evaluate the integrals in Exercises 31–78.
58. ∫(from 0 to ln9)e^θ(e^θ-1)^(1/2) dθ
Textbook Question
Evaluate the integrals in Exercises 31–78.
61. ∫(from 1 to 3)(ln(v+1))²/(v+1) dv
1
views