Properties of Definite Integrals

Definite integrals obey several properties that often make evaluation far shorter than direct antidifferentiation. Each is stated below with the situation it is used in.

P1 — The variable of integration is a dummy.

The value depends only on the function and the limits, not on the letter used for the variable.

P2 — Swapping the limits changes the sign.

P3 — Equal limits give zero.

The net area over an interval of zero width is zero.

P4 — Additivity over adjacent intervals. For any $c$,

An integral can be split at an intermediate point, or two adjacent integrals combined into one.

P5 — Reflection about the mid-point of the limits.

P6 — The special case $a = 0$.

This is P5 with lower limit $0$; it is the property most often used to evaluate integrals whose direct antiderivative is awkward, by adding the integral to a reflected copy of itself.

P7 — Symmetric limits and even/odd functions (developed fully in §5):

Illustration of P4. If $\displaystyle\int_{0}^{2} f(x)\,dx = 5$ and $\displaystyle\int_{2}^{6} f(x)\,dx = 3$, then $\displaystyle\int_{0}^{6} f(x)\,dx = 5 + 3 = 8$, without knowing $f$ itself.