# 定积分 ## 1 定积分性质 $$ \begin{aligned} & \mathop{ \int }\nolimits_{{a}}^{{b}}f{ \left( {x} \right) } \text{d} x=-\mathop{ \int }\nolimits_{{b}}^{{a}}f{ \left( {x} \right) } \text{d} x\\ & \mathop{ \int }\nolimits_{{a}}^{{b}}{ \left[ {f{ \left( {x} \right) } \pm g{ \left( {x} \right) }} \right] } \text{d} x=\mathop{ \int }\nolimits_{{a}}^{{b}}f{ \left( {x} \right) } \text{d} x \pm \mathop{ \int }\nolimits_{{a}}^{{b}}g{ \left( {x} \right) } \text{d} x\\ & \mathop{ \int }\nolimits_{{a}}^{{b}}kf{ \left( {x} \right) } \text{d} x=k\mathop{ \int }\nolimits_{{a}}^{{b}}f{ \left( {x} \right) } \text{d} x\\ & \mathop{ \int }\nolimits_{{a}}^{{b}}f{ \left( {x} \right) } \text{d} x=\mathop{ \int }\nolimits_{{a}}^{{c}}f{ \left( {x} \right) } \text{d} x+\mathop{ \int }\nolimits_{{c}}^{{b}}f{ \left( {x} \right) } \text{d} x, \forall c \in { \left( {a,b} \right) } \end{aligned} $$ ## 2 不等式 ### 一致大小 $$ \begin{aligned} & f{ \left( {x} \right) } \ge g{ \left( {x} \right) },x \in { \left[ {a,b} \right] } \Rightarrow \mathop{ \int }\nolimits_{{a}}^{{b}}f{ \left( {x} \right) } \text{d} x \ge \mathop{ \int }\nolimits_{{a}}^{{b}}g{ \left( {x} \right) } \text{d} x\\ \end{aligned} $$ ### 绝对值不等式 $$ \begin{aligned} & a < b \Rightarrow { \left| {\mathop{ \int }\nolimits_{{a}}^{{b}}f{ \left( {x} \right) } \text{d} x} \right| } \le \mathop{ \int }\nolimits_{{a}}^{{b}}{ \left| {f{ \left( {x} \right) } \text{d} x} \right| }\\ \end{aligned} $$ ### 上下限 $$ \begin{aligned} & {M=\mathop{{f}}\nolimits_{{max}}{ \left( {x} \right) },m=\mathop{{f}}\nolimits_{{min}}{ \left( {x} \right) },x \in { \left[ {a,b} \right] }}\\ & {m{ \left( {b-a} \right) } \le \mathop{ \int }\nolimits_{{a}}^{{b}}f{ \left( {x} \right) } \text{d} x \le M{ \left( {b-a} \right) }} \end{aligned} $$ ## 3 定积分定理 ### 牛顿莱布尼茨公式 $$ \begin{aligned} \mathop{ \int }\nolimits_{{a}}^{{b}}{F \prime }{ \left( {x} \right) } \text{d} x=F{ \left( {b} \right) }-F{ \left( {a} \right) } \end{aligned} $$ ### 积分中值定理1 $$ \begin{aligned} & {\text{若}\text{函}\text{数}\text{在}\text{闭}\text{区}\text{间}{ \left[ {a,b} \right] }\text{上}\text{连}\text{续}\text{,}\text{则}}\\ & { \exists \xi \in { \left[ {a,b} \right] }}\\ & {\mathop{ \int }\nolimits_{{a}}^{{b}}f{ \left( {x} \right) } \text{d} x=f{ \left( { \xi } \left) { \left( {b-a} \right) }\right. \right. }}\\ & {\text{若}f{ \left( {x} \right) }\text{和}g{ \left( {x} \right) }\text{在}\text{闭}\text{区}\text{间}{ \left[ {a,b} \right] }\text{上}\text{可}\text{积}\text{,}\text{且}g{ \left( {x} \right) }\text{在}\text{此}\text{区}\text{间}\text{上}\text{不}\text{变}\text{号}\text{,}\text{则}}\\ & {\mathop{ \int }\nolimits_{{a}}^{{b}}f{ \left( {x} \right) }g{ \left( {x} \right) } \text{d} x=f{ \left( { \xi } \right) }\mathop{ \int }\nolimits_{{a}}^{{b}}g{ \left( {x} \right) } \text{d} x} \end{aligned} $$ ### 积分中值定理2 $$ \begin{aligned} & {\text{若}f{ \left( {x} \right) }\text{和}g{ \left( {x} \right) }\text{在}\text{闭}\text{区}\text{间}{ \left[ {a,b} \right] }\text{上}\text{可}\text{积}\text{,}\text{且}f{ \left( {x} \right) }\text{为}\text{单}\text{调}\text{函}\text{数}\text{,}\text{则}}\\ & { \exists \xi \in { \left[ {a,b} \right] }}\\ & {\mathop{ \int }\nolimits_{{a}}^{{b}}f{ \left( {x} \right) }g{ \left( {x} \right) } \text{d} x=f{ \left( {a} \right) }\mathop{ \int }\nolimits_{{a}}^{{ \xi }}g{ \left( {x} \right) } \text{d} x+f{ \left( {b} \right) }\mathop{ \int }\nolimits_{{ \xi }}^{{b}}g{ \left( {x} \right) } \text{d} x}\\ & {\text{若}f{ \left( {x} \right) }\text{和}g{ \left( {x} \right) }\text{在}\text{闭}\text{区}\text{间}{ \left[ {a,b} \right] }\text{上}\text{可}\text{积}\text{,}f{ \left( {x} \right) } \ge 0\text{且}\text{为}\text{单}\text{调}\text{递}\text{减}\text{函}\text{数}\text{,}\text{则}}\\ & { \exists \xi \in { \left[ {a,b} \right] }}\\ & {\mathop{ \int }\nolimits_{{a}}^{{b}}f{ \left( {x} \right) }g{ \left( {x} \right) } \text{d} x=f{ \left( {a} \right) }\mathop{ \int }\nolimits_{{a}}^{{ \xi }}g{ \left( {x} \right) } \text{d} x}\\ & {\text{若}f{ \left( {x} \right) }\text{和}g{ \left( {x} \right) }\text{在}\text{闭}\text{区}\text{间}{ \left[ {a,b} \right] }\text{上}\text{可}\text{积}\text{,}f{ \left( {x} \right) } \ge 0\text{且}\text{为}\text{单}\text{调}\text{递}\text{增}\text{函}\text{数}\text{,}\text{则}}\\ & { \exists \xi \in { \left[ {a,b} \right] }}\\ & {\mathop{ \int }\nolimits_{{a}}^{{b}}f{ \left( {x} \right) }g{ \left( {x} \right) } \text{d} x=f{ \left( {b} \right) }\mathop{ \int }\nolimits_{{ \xi }}^{{b}}g{ \left( {x} \right) } \text{d} x} \end{aligned} $$ ### 变量替换公式 $$ \int_a^bf(x)dx=\int_\alpha^\beta f(\varphi(t))\varphi'(t)dt \\ \varphi(\alpha)=a,\varphi(\beta)=b $$
