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

3-4

Definite Integrals of Elementary Functions

3.0 Introduction

3.01 Theorems of a general nature

3.011 Suppose that f(x) is integrable^† over the largest of the intervals (p, q), (p, r), (r, q). Then (depending on the relative positions of the points p, q, and r) it is also integrable over the other two intervals and we have

${\int }_p^{q}f\left(x\right)\text{d}x={\int }_p^{r}f\left(x\right)\text{d}x+{\int }_r^{q}f\left(x\right)\text{d}x$

  FI II 126

3.012The first mean-value theorem. Suppose (1) that f(x) is continuous and that g(x) is integrable over the interval (p, q), (2) that m ≤ f(x) ≤M and (3) that g(x) does not change sign anywhere in the interval (p, q). Then, there exists at least one point ξ (with p ≤ ξ ≤ q) such ...