As regards integrable functions we have the following theorems:
1. If S and I are the superior and inferior limits (or greatest and least values) of ƒ(x) in the interval between a and b, ∫ ba ƒ(x)dx is intermediate between S(b − a) and I(b − a).
2. The integral is a continuous function of each of the end-values.
3. If the further end-value b is variable, and if ∫ xa ƒ(x)dx = F(x), then if ƒ(x) is continuous at b, F(x) is differentiable at b, and F′(b) = ƒ(b).
4. In case ƒ(x) is continuous throughout the interval F(x) is continuous and differentiable throughout the interval, and F′(x) = ƒ(x) throughout the interval.
5. In case ƒ′(x) is continuous throughout the interval between a and b,
∫ ba ƒ′(x)dx = ƒ(b) − ƒ(a).
6. In case ƒ(x) is discontinuous at one or more points of the interval between a and b, in which it is integrable,
∫ xa ƒ(x)dx
is a function of x, of which the four derivates at any point of the interval are equal to the limits of indefiniteness of ƒ(x) at the point.