Table of Integrals, Series, and Products, 8th Edition

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

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

  CA 105

with a ≤ ξ ≤ b.

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,

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

with a ≤ ξ ≤ b.

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