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], andfok.k ·f(x)dx=k. [*f(x)dr.aa(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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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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Question

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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

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