積分法
f ( x ) ≤ g ( x ) for ∀ x ∈ [ a , b ] ⇒ ∫ a b f ( x ) d x ≤ ∫ a b g ( x ) d x {\displaystyle f(x)\leq g(x){\mbox{ for }}\forall x\in [a,b]\Rightarrow \int _{a}^{b}f(x)dx\leq \int _{a}^{b}g(x)dx}
∫ a b f ′ ( x ) g ( x ) d x = [ f ( x ) g ( x ) ] a b − ∫ a b f ( x ) g ′ ( x ) d x . {\displaystyle \int _{a}^{b}f'(x)g(x)dx=[f(x)g(x)]_{a}^{b}-\int _{a}^{b}f(x)g'(x)dx.}
inf a ≤ x ≤ b { f ( x ) } ≤ 1 b − a ∫ a b f ( x ) d x ≤ sup a ≤ x ≤ b { f ( x ) } {\displaystyle \inf _{a\leq x\leq b}\{f(x)\}\leq {\frac {1}{b-a}}\int _{a}^{b}f(x)\,dx\leq \sup _{a\leq x\leq b}\{f(x)\}}