Use the Table of Integrals to evaluate the integral. dx S ex(5 + 10e* Select the correct answer. -X -e + 2 ln (e* + 2) 5 + C -5 e + 2ln (5 e + 2) -x -X 5 e +2 In (5 e + 2) -X -X -e + 2ln (5e +2) 25 -X 5 e + 2 In (5 e + C + C + 2) + C
Use the Table of Integrals to evaluate the integral. dx S ex(5 + 10e* Select the correct answer. -X -e + 2 ln (e* + 2) 5 + C -5 e + 2ln (5 e + 2) -x -X 5 e +2 In (5 e + 2) -X -X -e + 2ln (5e +2) 25 -X 5 e + 2 In (5 e + C + C + 2) + C
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
![**Integral Evaluation Using the Table of Integrals**
In this exercise, we are asked to use the Table of Integrals to evaluate the given integral. The integral to be evaluated is:
\[ \int \frac{dx}{e^{-x}(5 + 10e^{-x})} \]
**Step-by-Step Solution and Answer Choices**
Let's explore the provided answer choices to identify the correct solution to the integral:
1. \(\frac{-e^{-x} + 2 \ln(e^{-x} + 2)}{5} + C\)
2. \(\frac{-5e^{-x} + 2 \ln(e^{-x} + 2)}{5} + C\)
3. \(-5e^{-x} + 2 \ln(e^{-x} + 2) + C\)
4. \(\frac{-e^{-x} + 2 \ln(5e^{-x} + 2)}{25} + C\)
5. \(-5e^{-x} + 2 \ln(5e^{-x} + 2) + C\)
The correct answer will be chosen from one of these options. Let's break down the steps to solve this integral:
**Solution Steps:**
1. **Simplify the Integral:**
Simplify the integrand initially:
\[ \int \frac{dx}{e^{-x}(5 + 10e^{-x})} = \int \frac{1}{e^{-x}(5 + 10e^{-x})} dx \]
\[ = \int \frac{1}{5e^{-x} + 10e^{-2x}} dx \]
2. **Substitution:**
Use a suitable substitution to simplify the integral. Let \(u = e^{-x}\):
\[ du = -e^{-x} dx \]
\[ dx = -\frac{du}{u} \]
3. **Transform the Integral:**
Substituting the variables into the integral:
\[ \int \frac{1}{5u + 10u^2} \cdot -\frac{du}{u} \]
\[ = -\int \frac{1}{u(5u + 10u^2)} du \]
\[ = -\int \frac{1}{5u^2 + 10u^3} du \]
Continuing, solve the integral using partial fraction decomposition or reference](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F24e8bfde-46ac-4815-81bc-d8be1d9c4f35%2Fda46ffa2-75ee-4f25-a144-4cc17ec62dda%2Fvsfhp4s_processed.png&w=3840&q=75)
Transcribed Image Text:**Integral Evaluation Using the Table of Integrals**
In this exercise, we are asked to use the Table of Integrals to evaluate the given integral. The integral to be evaluated is:
\[ \int \frac{dx}{e^{-x}(5 + 10e^{-x})} \]
**Step-by-Step Solution and Answer Choices**
Let's explore the provided answer choices to identify the correct solution to the integral:
1. \(\frac{-e^{-x} + 2 \ln(e^{-x} + 2)}{5} + C\)
2. \(\frac{-5e^{-x} + 2 \ln(e^{-x} + 2)}{5} + C\)
3. \(-5e^{-x} + 2 \ln(e^{-x} + 2) + C\)
4. \(\frac{-e^{-x} + 2 \ln(5e^{-x} + 2)}{25} + C\)
5. \(-5e^{-x} + 2 \ln(5e^{-x} + 2) + C\)
The correct answer will be chosen from one of these options. Let's break down the steps to solve this integral:
**Solution Steps:**
1. **Simplify the Integral:**
Simplify the integrand initially:
\[ \int \frac{dx}{e^{-x}(5 + 10e^{-x})} = \int \frac{1}{e^{-x}(5 + 10e^{-x})} dx \]
\[ = \int \frac{1}{5e^{-x} + 10e^{-2x}} dx \]
2. **Substitution:**
Use a suitable substitution to simplify the integral. Let \(u = e^{-x}\):
\[ du = -e^{-x} dx \]
\[ dx = -\frac{du}{u} \]
3. **Transform the Integral:**
Substituting the variables into the integral:
\[ \int \frac{1}{5u + 10u^2} \cdot -\frac{du}{u} \]
\[ = -\int \frac{1}{u(5u + 10u^2)} du \]
\[ = -\int \frac{1}{5u^2 + 10u^3} du \]
Continuing, solve the integral using partial fraction decomposition or reference
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 4 steps with 4 images
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