definite integral

if $f(x)$ is continuous on $[a,b]$ and $F(x)$ is the anti-derivative of $f(x)$ with respect to $x$, then

${\int }_a^{b}f(x)dx=F(x){|}_a^b=F(b)-F(a)$

Rules

• Order of integration: ${\int }_a^{b}f(x)dx=-{\int }_b^{a}f(x)dx$
• Zero width: ${\int }_a^{a}f(x)dx=0$
• Constant multiple: ${\int }_a^{b}kf(x)dx=k{\int }_a^{b}f(x)dx$
• Sum/difference: ${\int }_a^b(f(x)\pm g(x))dx={\int }_a^{b}f(x)dx\pm {\int }_a^{b}g(x)dx$
• Additivity: ${\int }_a^{b}f(x)dx+{\int }_b^{c}f(x)dx={\int }_a^{c}f(x)dx$
• Max-min Inequality: If there exists f-max and f-min on $[a,b]$, then $f_{min}(b-a)\le {\int }_a^{b}f(x)dx\le f_{max}(b-a)$
• Domination: if $f(x)\ge g(x)$ on $[a,b]$, then ${\int }_a^{b}f(x)dx\ge {\int }_a^{b}g(x)dx$