Some Properties of Definite Integrals

Definite integrals obey a set of properties that make many integrals far easier to evaluate than working from the fundamental theorem directly, especially when the integrand has some underlying symmetry.

• Independence of the variable name: $\displaystyle\int_a^b f(x)\,dx = \int_a^b f(t)\,dt$ — the letter used for the variable of integration does not affect the value, since both reduce to $F(b)-F(a)$.
• Reversing the limits flips the sign: $\displaystyle\int_a^b f(x)\,dx = -\int_b^a f(x)\,dx$.
• Splitting the interval: for any point $c$, $\displaystyle\int_a^b f(x)\,dx = \int_a^c f(x)\,dx + \int_c^b f(x)\,dx$.
• Reflecting about the midpoint: $\displaystyle\int_a^b f(x)\,dx = \int_a^b f(a+b-x)\,dx$, obtained by substituting $t=a+b-x$.
• A special case of the above, with $a=0$: $\displaystyle\int_0^a f(x)\,dx = \int_0^a f(a-x)\,dx$.
• Doubling the interval: $\displaystyle\int_0^{2a} f(x)\,dx = \int_0^a f(x)\,dx + \int_0^a f(2a-x)\,dx$.

Combining the last property with the sign-flip property gives a useful consequence for behaviour symmetric about $x=a$:

The most frequently used special case involves even and odd functions. Recall $f$ is even if $f(-x)=f(x)$ (e.g. $f(x)=x^2$) and odd if $f(-x)=-f(x)$ (e.g. $f(x)=x^3$). Then: …