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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

${\int}_{a}^{b}f\left(x\right)g\left(x\right)\text{d}x}=f\left(\xi \right){\displaystyle {\int}_{a}^{b}g\left(x\right)}\text{d}x,$

CA 105

with a ≤ ξ ≤ b.

(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,

${\int}_{a}^{b}f\left(x\right)g\left(x\right)\text{d}x}=f\left(a\right){\displaystyle {\int}_{a}^{\xi}g\left(x\right)}\text{d}x,$

with a ≤ ξ ≤ b.

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

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