Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
Related questions
Question
![**Integration Problem: Solving a Definite Integral**
Let's evaluate the following definite integral:
\[ \int_{0}^{\pi/12} \frac{1}{\sqrt{1 - \cos{12x}}} \, dx \]
---
**Detailed Steps:**
We are given the integral:
\[ \int_{0}^{\pi/12} \frac{1}{\sqrt{1 - \cos{12x}}} \, dx \]
To solve this integral, one might need to use trigonometric identities or substitutions to simplify the integrand. One common approach is to use the identity for the cosine of multiple angles or converting the expression under the square root into a more manageable form. Examples of strategies include:
1. **Using Trigonometric Identities:**
- \( \cos{2\theta} = 1 - 2\sin^2{\theta} \)
- \( 1 - \cos{2\theta} = 2\sin^2{\theta} \)
2. **Substitution Method:**
- Setting \( u = 12x \)
- Transform the limits of integration accordingly.
After employing appropriate transformations, simplify the integral and solve it step-by-step.
If you are familiar with advanced integration techniques or if the integral corresponds to a known integral table, you might find a direct solution more efficiently.
*Note: Make sure to simplify the final expression and substitute back if you've used any substitution.*
\[ \int_{0}^{\pi/12} \frac{1}{\sqrt{1 - \cos{12x}}} \, dx = \boxed{} \]
(In the box, provide the final simplified solution.)
---
This problem involves evaluating a definite integral with a trigonometric function under a square root. Such integrals frequently appear in problems related to electrical engineering, physics, and more advanced calculus.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1c5e517d-57db-4b65-baa5-8a292d97b7f5%2Fc2d30f54-e6c3-4d98-b45a-9b334ead2045%2Fs9cm9qs_processed.png&w=3840&q=75)
Transcribed Image Text:**Integration Problem: Solving a Definite Integral**
Let's evaluate the following definite integral:
\[ \int_{0}^{\pi/12} \frac{1}{\sqrt{1 - \cos{12x}}} \, dx \]
---
**Detailed Steps:**
We are given the integral:
\[ \int_{0}^{\pi/12} \frac{1}{\sqrt{1 - \cos{12x}}} \, dx \]
To solve this integral, one might need to use trigonometric identities or substitutions to simplify the integrand. One common approach is to use the identity for the cosine of multiple angles or converting the expression under the square root into a more manageable form. Examples of strategies include:
1. **Using Trigonometric Identities:**
- \( \cos{2\theta} = 1 - 2\sin^2{\theta} \)
- \( 1 - \cos{2\theta} = 2\sin^2{\theta} \)
2. **Substitution Method:**
- Setting \( u = 12x \)
- Transform the limits of integration accordingly.
After employing appropriate transformations, simplify the integral and solve it step-by-step.
If you are familiar with advanced integration techniques or if the integral corresponds to a known integral table, you might find a direct solution more efficiently.
*Note: Make sure to simplify the final expression and substitute back if you've used any substitution.*
\[ \int_{0}^{\pi/12} \frac{1}{\sqrt{1 - \cos{12x}}} \, dx = \boxed{} \]
(In the box, provide the final simplified solution.)
---
This problem involves evaluating a definite integral with a trigonometric function under a square root. Such integrals frequently appear in problems related to electrical engineering, physics, and more advanced calculus.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 3 steps
Recommended textbooks for you
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus
ISBN:
9781319050740
Author:
Jon Rogawski, Colin Adams, Robert Franzosa
Publisher:
W. H. Freeman
Calculus: Early Transcendental Functions
Calculus
ISBN:
9781337552516
Author:
Ron Larson, Bruce H. Edwards
Publisher:
Cengage Learning