Properties of the definite integral.

Proposition 8.4.1   $\int_a^a f(x) \; dx =0$ .

Proposition 8.4.2   Let $f$ be a function integrable on an interval $I$ and let $a,b,c \in I$ . Then:

$$\int_a^b f(x) \; dx = \int_a^c f(x) \; dx + \int_c^b f(x) \; dx.$$

Proposition 8.4.3   Suppose that the functions $f$ and $g$ are integrable on $[a,b]$ . Then $f+g$ is integrable on $[a,b]$ and:

$$\int_a^b [f(x)+g(x)] \; dx = \int_a^b f(x) \; dx + \int_a^b g(x) \; dx.$$

Proposition 8.4.4   Suppose that the function $f$ is integrable on $[a,b]$ . Then $\alpha f$ is integrable on $[a,b]$ and:

$$\int_a^b \alpha f(x) \; dx = \alpha \int_a^b f(x) \; dx.$$

Proposition 8.4.5   If $f$ is integrable on $[a,b]$ and if $f(x) \geq 0$ on $[a,b]$ , then $\int_a^b f(x) \; dx \geq 0$ .

Corollary 8.4.6   If $f$ and $g$ are integrable on $[a,b]$ and if $f(x) \geq g(x)$ on $[a,b]$ , then $\int_a^b f(x) \; dx \geq \int_a^b g(x) \; dx$ .

Corollary 8.4.7   If $f$ is integrable on $[a,b]$ , then $\vert f\vert$ is also integrable on $[a,b]$ and $\left\vert \int_a^b f(x) \; dx \right\vert \leq \int_a^b \vert f(x)\vert \; dx$ .

For definite integrals, the method of substitution (v.s. 2.2) is described as follows:

Theorem 8.4.8   Let $a,b$ be two numbers such that $a<b$ Let $g$ be a function having a continuous first derivative on $[a,b]$ and such that $a \leq x \leq b \Longrightarrow m \leq g(t) \leq M$ . If $f$ is a function, continuous on the interval $[m,M]$ , then for any $u \in [a,b]$ we have:

$$\int_{g(a)}^{g(u)} f(t) \; dt = \int_a^u f[g(x)] \cdot g'(x) \; dx$$

In particular

$$\int_{g(a)}^{g(b)} f(t) \; dt = \int_a^b f[g(x)] \cdot g'(x) \; dx$$