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5. Use a power series to approximate the definite integral to 6 decimal places. 0.1 a. 50.211 dx b. fo.¹ x arctan 3x dx 1+x5

5. Use a power series to approximate the definite integral to 6 decimal places. 0.1 a. 50.211 dx b. fo.¹ x arctan 3x dx 1+x5

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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# Approximating Definite Integrals Using Power Series ## Objective: Learn how to use power series to approximate definite integrals to six decimal places. ## Problem: 5. **Use a power series to approximate the definite integral to 6 decimal places.** a. \( \int_0^{0.2} \frac{1}{1 + x^5} \, dx \) b. \( \int_0^{0.1} x \arctan(3x) \, dx \) --- ## Explanation: To solve these integrals using power series, we expand each function into a series representation, integrate term by term, and evaluate the result to obtain an approximation. ### Steps: 1. **Expand the function into a Power Series:** - Identify the form of the function that allows expansion into a power series. 2. **Integrate Term by Term:** - Substitute the power series into the definite integral and perform the integration on each term. 3. **Evaluate the Integral:** - Summation of the series terms evaluated at the given bounds will yield an approximation of the definite integral. 4. **Round the Result:** - Ensure the final value is rounded to six decimal places for accuracy. By following these steps, you can efficiently approximate definite integrals even when analytical solutions are complex or unreachable. --- This exercise develops skills in combining series knowledge with integration techniques, valuable for tackling complex calculus problems.
Transcribed Image Text:# Approximating Definite Integrals Using Power Series ## Objective: Learn how to use power series to approximate definite integrals to six decimal places. ## Problem: 5. **Use a power series to approximate the definite integral to 6 decimal places.** a. \( \int_0^{0.2} \frac{1}{1 + x^5} \, dx \) b. \( \int_0^{0.1} x \arctan(3x) \, dx \) --- ## Explanation: To solve these integrals using power series, we expand each function into a series representation, integrate term by term, and evaluate the result to obtain an approximation. ### Steps: 1. **Expand the function into a Power Series:** - Identify the form of the function that allows expansion into a power series. 2. **Integrate Term by Term:** - Substitute the power series into the definite integral and perform the integration on each term. 3. **Evaluate the Integral:** - Summation of the series terms evaluated at the given bounds will yield an approximation of the definite integral. 4. **Round the Result:** - Ensure the final value is rounded to six decimal places for accuracy. By following these steps, you can efficiently approximate definite integrals even when analytical solutions are complex or unreachable. --- This exercise develops skills in combining series knowledge with integration techniques, valuable for tackling complex calculus problems.
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