Evaluate the integral below by first doing a u substitution x = xa 1+x² xa the de integral is done in the section of the textbook called "An indentation around a branch point.") (1 + x²)² as the 0 de integral was done in class. (This in turn is done quite similarly as /0 ∞ -1/2 x +9 u² and then arguing X dx
Evaluate the integral below by first doing a u substitution x = xa 1+x² xa the de integral is done in the section of the textbook called "An indentation around a branch point.") (1 + x²)² as the 0 de integral was done in class. (This in turn is done quite similarly as /0 ∞ -1/2 x +9 u² and then arguing X dx
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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![### Problem Statement
Evaluate the integral below by first doing a \( u \) substitution \( x = u^2 \) and then arguing as the \( \int_{0}^{\infty} \frac{x^a}{1 + x^2} \, dx \) integral was done in class. (This in turn is done quite similarly as the \( \int_{0}^{\infty} \frac{x^a}{(1 + x^2)^2} \, dx \) integral is done in the section of the textbook called "An indentation around a branch point.")
\[
\int_{0}^{\infty} \frac{x^{-\frac{1}{3}}}{x + 9} \, dx
\]
### Explanation
- Begin by substituting \( x = u^2 \) in the given integral.
- Analyze the similarities to previously solved integrals \( \int_{0}^{\infty} \frac{x^a}{1 + x^2} \, dx \) and \( \int_{0}^{\infty} \frac{x^a}{(1 + x^2)^2} \, dx \).
- Refer to the section "An indentation around a branch point" in your textbook for detailed methods to evaluate similar types of integrals.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa62f7b35-7db6-46d8-92c3-a45ad2747ea7%2F875b02dd-8875-431c-a577-1344fc0e9e5c%2F9s7wkjj_processed.png&w=3840&q=75)
Transcribed Image Text:### Problem Statement
Evaluate the integral below by first doing a \( u \) substitution \( x = u^2 \) and then arguing as the \( \int_{0}^{\infty} \frac{x^a}{1 + x^2} \, dx \) integral was done in class. (This in turn is done quite similarly as the \( \int_{0}^{\infty} \frac{x^a}{(1 + x^2)^2} \, dx \) integral is done in the section of the textbook called "An indentation around a branch point.")
\[
\int_{0}^{\infty} \frac{x^{-\frac{1}{3}}}{x + 9} \, dx
\]
### Explanation
- Begin by substituting \( x = u^2 \) in the given integral.
- Analyze the similarities to previously solved integrals \( \int_{0}^{\infty} \frac{x^a}{1 + x^2} \, dx \) and \( \int_{0}^{\infty} \frac{x^a}{(1 + x^2)^2} \, dx \).
- Refer to the section "An indentation around a branch point" in your textbook for detailed methods to evaluate similar types of integrals.
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