PROPOSITION 7.17. Suppose that ƒ, g : [a, b] → R are integrable. Then: (1) For any k € R, k. f is also integrable on [a, b], and ·b ·b S k. f(x) dx = k [ f(x) dx. a a (2) The function f + g is also integrable on [a, b], and [*(ƒ + 9) (x) dx = [*ƒ(x) dx + [*9(x)dx.
PROPOSITION 7.17. Suppose that ƒ, g : [a, b] → R are integrable. Then: (1) For any k € R, k. f is also integrable on [a, b], and ·b ·b S k. f(x) dx = k [ f(x) dx. a a (2) The function f + g is also integrable on [a, b], and [*(ƒ + 9) (x) dx = [*ƒ(x) dx + [*9(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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Can you please simply prove both propositions (1 & 2) from the attached note?
For proposition (1):
Please prove it in the case that k < 0 or k = 0.
Thank you!
![PROPOSITION 7.17. Suppose that f, g : [a, b] → R are integrable. Then:
(1) For any k = R, kf is also integrable on [a, b], and
fok.
k ·
f(x)dx=k. [*f(x)dr.
a
a
(2) The function f + g is also integrable on [a, b], and
[^(f + 9){(z)dx = [[ f(x)dz + [*9(z)dx.
a](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fdb708fa5-116d-42c3-bb62-31dd00678e29%2Fe6b455a6-e012-4ff4-81a3-8a112516395d%2F5e2l4w_processed.png&w=3840&q=75)
Transcribed Image Text:PROPOSITION 7.17. Suppose that f, g : [a, b] → R are integrable. Then:
(1) For any k = R, kf is also integrable on [a, b], and
fok.
k ·
f(x)dx=k. [*f(x)dr.
a
a
(2) The function f + g is also integrable on [a, b], and
[^(f + 9){(z)dx = [[ f(x)dz + [*9(z)dx.
a
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