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Table of Integrals, Series, and Products, 8th Edition by Daniel Zwillinger

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11

Integral Inequalities

11.11 Mean Value Theorems

11.111 First mean value theorem

Let f(x) and g(x) be two bounded functions integrable in [a, b] and let g(x) be of one sign in this interval. Then

abf(x)g(x)dx=f(ξ)abg(x)dx,

si1_e  CA 105

with a ≤ ξ ≤ b.

11.112 Second mean value theorem

(i)  Let f(x) be a bounded, monotonic decreasing, and nonnegative function in [a, b], and let g(x) be a bounded integrable function. Then,

abf(x)g(x)dx=f(a)aξg(x)dx,

si2_e

with a ≤ ξ ≤ b.

(ii)  Let f(x) be a bounded, monotonic increasing, and nonnegative function ...

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