3. Prove properties (3) and (4) analytically, by making x = -t in the part of the Integral from -a to 0 and using the definitions (1) and (2).
3. Prove properties (3) and (4) analytically, by making x = -t in the part of the Integral from -a to 0 and using the definitions (1) and (2).
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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Riemann Integral
Bernhard Riemann's integral was the first systematic description of the integral of a function on an interval in the branch of mathematics known as real analysis.
Question
Transcribed Image Text:3. Prove properties (3) and (4) analytically, by making x = -t in the part of
the Integral from -a to 0 and using the definitions (1) and (2).
Transcribed Image Text:val) is said to be even if
f(-x) = f(x),
(1)
(x) is said to be odd if
f(-x) = -f(x).
(2)
xample, x² and cos x are even, and x and sin x are odd. The graph of
ren function is symmetric about the y-axis, as shown in Figure 48, and
raph of an odd function is skew-symmetric (Figure 49). By putting x = 0
,we see that an odd function always has the property that f (0) = 0. It is
from the figures that
[f(x)dx = 2[f(x)dx
if f(x)is even,
(3)
-a
(x)dx = 0 if f(x)is odd,
(4)
-a
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