Applying Properties of Definite Integrals AB & BC

Definite integrals obey a short list of rules that need no knowledge of the integrand. Zero width gives $\int_{a}^{a} f = 0$; reversal gives $\int_{b}^{a} f = -\int_{a}^{b} f$, so swapping the limits negates rather than leaving the value alone; and additivity gives $\int_{a}^{b} f = \int_{a}^{c} f + \int_{c}^{b} f$ for any $c$, which also lets you subtract a shorter run from a longer one. Linearity adds two more: constant multiples come out front and sums split apart, and those are what make a table of given values enough to answer with.

What is absent from the list matters as much. $\int fg$ is not $\left(\int f\right)\left(\int g\right)$: on $[0,2]$, $\int x\,dx = 2$ twice over, while $\int x \cdot x\,dx = \frac{8}{3}$, and $2 \times 2 = 4$. Quotients, composites and powers fail for the same reason, since integration distributes over addition and nothing more, and the constant in constant-multiple must be a number rather than an expression in the variable. On a centred interval, an even integrand doubles the half from $0$ to $a$ and an odd one integrates to zero, which is a cancellation of signed areas and not an absence of region.