Properties of Definite Integrals

Properties of the Definite Integral

1. $\boxed{\displaystyle \int_{a}^{a}f(x)\text{d}x=0}$
2. $\displaystyle \boxed{\int_{a}^{b}k\text{d}x=k(b-a)}$ for any constant $k$
3. $\displaystyle \boxed{\int_{a}^{b}f(x)\text{d}x=\int_{a}^{c}f(x)\text{d}x+\int_{c}^{b}f(x)\text{d}x}$ for all $c$ in $[a,\ b]$
4. $\boxed{\displaystyle \int_{a}^{b}[f(x)\pm g(x)]\text{d}x=\int_{a}^{b}f(x)\text{d}x\pm\int_{a}^{b}g(x)\text{d}x}$
5. $\displaystyle \boxed{\int_{a}^{b}kf(x)\text{d}x=k\int_{a}^{b}f(x)\text{d}x}$ for any constant $k$
6. $\boxed{\displaystyle \int_{a}^{b}f(x)\text{d}x=-\int_{b}^{a}f(x)\text{d}x}$
7. $\boxed{\displaystyle\int_{-a}^af\left(x\right)\text{ d}x=0 \text{, \ \ if } f(x) \text{ is an odd function}}$
8. $\boxed{\displaystyle \int_{-a}^{a}f(x)\text{ d}x=2\int_0^af(x)\text{ d}x \text{, \ \ if } f(x) \text{ is an even function}}$

Mean Value Theorem for Definite Integrals