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9.12 Let f(x) be a function co each of intervals (a, c₁), (C₁,

9.12 Let f(x) be a function co each of intervals (a, c₁), (C₁,

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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Theorem 9.12 Let f(x) be a function continuous on [a, b] and let g(x) be constant in each of intervals (a, c₁), (C₁, C2),.........., (Cm, b) where a<c₁ <C₂ <.....< Cm <b i.e. g(x) is a step function. Then b [ f(x) dg(x) = f(a) [g(a + 0) - g(a)] a m + Σf(ck) [8(ck + 0) - g(ck - 0)] + f(b) [g(b)-g(b-0)] k=0
Transcribed Image Text:Theorem 9.12 Let f(x) be a function continuous on [a, b] and let g(x) be constant in each of intervals (a, c₁), (C₁, C2),.........., (Cm, b) where a<c₁ <C₂ <.....< Cm <b i.e. g(x) is a step function. Then b [ f(x) dg(x) = f(a) [g(a + 0) - g(a)] a m + Σf(ck) [8(ck + 0) - g(ck - 0)] + f(b) [g(b)-g(b-0)] k=0
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