2. Two problems: (a) Prove the formula for integration by parts: if F and G are C¹ functions on [a, b] and f = F', 8 = G', then [ "FG - ſ³ ƒ(x)G(x) dx. F(x)g (x) dx = F(b)G(b)-F(a)G(a)-
2. Two problems: (a) Prove the formula for integration by parts: if F and G are C¹ functions on [a, b] and f = F', 8 = G', then [ "FG - ſ³ ƒ(x)G(x) dx. F(x)g (x) dx = F(b)G(b)-F(a)G(a)-
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 2: Two Problems
**(a)** Prove the formula for *integration by parts*: If \( F \) and \( G \) are \( C^1 \) functions on \([a,b]\) and \( f = F' \), \( g = G' \), then
\[
\int_a^b F(x)g(x) \, dx = F(b)G(b) - F(a)G(a) - \int_a^b f(x)G(x) \, dx.
\]
**(b)** Compute \( a_n = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} x^n e^{-x^2/2} \, dx = \lim_{L \to \infty} \left( \frac{1}{\sqrt{2\pi}} \int_0^L x^n e^{-x^2/2} \, dx \right) + \lim_{L \to \infty} \left( \frac{1}{\sqrt{2\pi}} \int_L^0 x^n e^{-x^2/2} \, dx \right).
You may take for granted that \( a_0 = 1 \); we'll prove this in a future section.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F77cfc5ac-076f-4cb7-b69f-6b7f1cfee42f%2Ff134ae55-8ad3-4bd6-8e8b-d1c81d5ccf5a%2Fnye71i9_processed.png&w=3840&q=75)
Transcribed Image Text:### Problem 2: Two Problems
**(a)** Prove the formula for *integration by parts*: If \( F \) and \( G \) are \( C^1 \) functions on \([a,b]\) and \( f = F' \), \( g = G' \), then
\[
\int_a^b F(x)g(x) \, dx = F(b)G(b) - F(a)G(a) - \int_a^b f(x)G(x) \, dx.
\]
**(b)** Compute \( a_n = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} x^n e^{-x^2/2} \, dx = \lim_{L \to \infty} \left( \frac{1}{\sqrt{2\pi}} \int_0^L x^n e^{-x^2/2} \, dx \right) + \lim_{L \to \infty} \left( \frac{1}{\sqrt{2\pi}} \int_L^0 x^n e^{-x^2/2} \, dx \right).
You may take for granted that \( a_0 = 1 \); we'll prove this in a future section.
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