#Properties of the definite integral The definite integral has several different properties that you should know. Some of these properties allow you to easily integrate simple functions, and some are just interesting to know. Others can be applied to problems. In this note set I am not really going to give a detailed explanation of each one, I am rather just going to show these properties. ##The properties 1. If c is a real number then: \[ \int^{b}_{a}cdx = c(b-a) \] 2. If \(f\) is integrable on \([a,b]\) and \(c\) is a any real number then \(cf\) is integrable on \([a,b]\) and \[ \int^{b}_{a}cf(x)dx = c \int^{b}_{a}f(x)dx \] 3. If \(g\) and \(f\) are integrable on \([a,b]\) then \(g+f\) and \(f-g\) are integrable on \([a,b]\) and \[ \int^{b}_{a}[f(x) + g(x)]dx= \int^{b}_{a}f(x)dx + \int^{b}_{a}g(x)dx \] \[ \int^{b}_{a}[f(x) - g(x)]dx = \int^{b}_{a}f(x)dx - \int^{b}_{a}g(x)dx \] 4. If \(f\) is integrable on \([a,c]\) and \([c,b]\) and \(a\lt c\lt b\) then \(f\) is integrable on \([a,b]\) and \[ \int_{a}^{b}f(x)dx=\int_{a}^{c}f(x)dx+\int^{b}_{c}f(x)dx \] 5. If \(f\) is integrable on a closed interval and if \(a,b,c\) are any three numbers in the interval then: \[ \int_{a}^{b}f(x)dx = \int_{a}^{c}f(x)dx+\int_{c}^{b}f(x)dx \] 6. If \(f(x) \geq 0\) for every \(x\) in \([a,b]\) and \(f(x)\) is integrable on \([a,b]\) then: \[ \int_{a}^{b}f(x)dx\geq0 \] 7. If \(f\) and \(g\) are integrable on the closed interval \([a,b]\) and \(f(x) \geq g(x)\) for every \(x\) in the closed interval \([a,b]\) then: \[ \int_{a}^{b}f(x)dx \geq \int_{a}^{b}g(x)dx \] 8. The mean value thereom for definite integrals: If \(f\) is continuous on a closed interval \([a,b]\) then there is a number \(z\) in the open interval \((a,b)\) such that: \[ \int_{a}^{b}f(x)dx=f(z)(b-a) \] 9. If \(f\) is continuous on \([a,b]\) the average value \(f_{av}\) of \(f\) on \([a,b]\) is: \[ f_{av} = \frac{1}{b-a}\int_{a}^{b}f(x)dx \] ##Conclusion Some of these properties are quite interesting. I find the last one quite interesting. However, while not all of them need to be memorized, you should use a page like this in order to refer to some rules that you may need in the future.