Express the definite integral as an infinite series. 7e dr
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![**Task: Express the definite integral as an infinite series.**
\[ \int_0^1 7e^{-x^3} \, dx \]
This expression represents the definite integral of the function \(7e^{-x^3}\) from 0 to 1. To express this integral as an infinite series, one approach is to use the series expansion of the exponential function \(e^{-x^3}\).
### Steps to Express as Series:
1. **Expand \(e^{-x^3}\) as a Power Series:**
The exponential function \(e^{-x^3}\) can be expanded using the Taylor series:
\[ e^{-x^3} = \sum_{n=0}^{\infty} \frac{(-x^3)^n}{n!} = \sum_{n=0}^{\infty} \frac{(-1)^n x^{3n}}{n!} \]
2. **Multiply by the Constant:**
Since the integrand is \(7e^{-x^3}\), multiply the series by 7:
\[ 7 \cdot e^{-x^3} = \sum_{n=0}^{\infty} \frac{7(-1)^n x^{3n}}{n!} \]
3. **Integrate Term by Term:**
Now, integrate term by term from 0 to 1:
\[ \int_0^1 7e^{-x^3} \, dx = \sum_{n=0}^{\infty} \int_0^1 \frac{7(-1)^n x^{3n}}{n!} \, dx \]
Evaluate each integral:
\[ \int_0^1 x^{3n} \, dx = \left[\frac{x^{3n+1}}{3n+1}\right]_0^1 = \frac{1}{3n+1} \]
4. **Resulting Series:**
The definite integral expressed as an infinite series is:
\[ \sum_{n=0}^{\infty} \frac{7(-1)^n}{n!(3n+1)} \]
This expansion allows for the approximation of the integral to any desired accuracy by taking a sufficient number of terms from the series.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc38b55f7-e4f6-4386-af07-c4ae4df48ed0%2F89800854-22a2-450f-acdc-a8bf7b427e63%2F6poenwm_processed.png&w=3840&q=75)
Transcribed Image Text:**Task: Express the definite integral as an infinite series.**
\[ \int_0^1 7e^{-x^3} \, dx \]
This expression represents the definite integral of the function \(7e^{-x^3}\) from 0 to 1. To express this integral as an infinite series, one approach is to use the series expansion of the exponential function \(e^{-x^3}\).
### Steps to Express as Series:
1. **Expand \(e^{-x^3}\) as a Power Series:**
The exponential function \(e^{-x^3}\) can be expanded using the Taylor series:
\[ e^{-x^3} = \sum_{n=0}^{\infty} \frac{(-x^3)^n}{n!} = \sum_{n=0}^{\infty} \frac{(-1)^n x^{3n}}{n!} \]
2. **Multiply by the Constant:**
Since the integrand is \(7e^{-x^3}\), multiply the series by 7:
\[ 7 \cdot e^{-x^3} = \sum_{n=0}^{\infty} \frac{7(-1)^n x^{3n}}{n!} \]
3. **Integrate Term by Term:**
Now, integrate term by term from 0 to 1:
\[ \int_0^1 7e^{-x^3} \, dx = \sum_{n=0}^{\infty} \int_0^1 \frac{7(-1)^n x^{3n}}{n!} \, dx \]
Evaluate each integral:
\[ \int_0^1 x^{3n} \, dx = \left[\frac{x^{3n+1}}{3n+1}\right]_0^1 = \frac{1}{3n+1} \]
4. **Resulting Series:**
The definite integral expressed as an infinite series is:
\[ \sum_{n=0}^{\infty} \frac{7(-1)^n}{n!(3n+1)} \]
This expansion allows for the approximation of the integral to any desired accuracy by taking a sufficient number of terms from the series.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 2 steps with 1 images
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
