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Show that dx (a) - x² +1 2

Show that dx (a) - x² +1 2

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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The text reads: "#18(a), (b), (d), page 169. For part (d) you may assume that \( a^2 \neq b^2 \)."
Transcribed Image Text:The text reads: "#18(a), (b), (d), page 169. For part (d) you may assume that \( a^2 \neq b^2 \)."
**18. Show that** (a) \(\int_{0}^{\infty} \frac{dx}{x^2 + 1} = \frac{\pi}{2},\) (b) \(\int_{-\infty}^{\infty} \frac{x^2 + 1}{x^4 + 1} \, dx = \sqrt{2\pi},\) (c) \(\int_{0}^{\infty} \frac{dx}{(x^2 + 1)^2} = \frac{\pi}{4},\) (d) \(\int_{0}^{\infty} \frac{ab}{(x^2 + a^2)(x^2 + b^2)} \, dx = \frac{\pi}{2(a + b)}.\)
Transcribed Image Text:**18. Show that** (a) \(\int_{0}^{\infty} \frac{dx}{x^2 + 1} = \frac{\pi}{2},\) (b) \(\int_{-\infty}^{\infty} \frac{x^2 + 1}{x^4 + 1} \, dx = \sqrt{2\pi},\) (c) \(\int_{0}^{\infty} \frac{dx}{(x^2 + 1)^2} = \frac{\pi}{4},\) (d) \(\int_{0}^{\infty} \frac{ab}{(x^2 + a^2)(x^2 + b^2)} \, dx = \frac{\pi}{2(a + b)}.\)
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