# 常见积分公式 ## 1 基本公式 $$ \begin{aligned} &{{}_{ }^{ } \int _{ }^{ }k \text{d} x=kx+C}\\ &{{}_{ }^{ } \int _{ }^{ }\mathop{{x}}\nolimits^{{ \mu }} \text{d} x=\frac{{\mathop{{x}}\nolimits^{{ \mu +1}}}}{{ \mu +1}}+C,{ \left( { \mu \neq -1} \right) }}\\ &{{}_{ }^{ } \int _{ }^{ }\frac{{1}}{{x}} \text{d} x= \text{ln} { \left| {x} \right| }+C}\\ & {{}_{ }^{ } \int _{ }^{ }\frac{{1}}{{1+\mathop{{x}}\nolimits^{{2}}}} \text{d} x= \text{arctan} x+C}\\ &{{}_{ }^{ } \int _{ }^{ }\frac{{1}}{{\sqrt{{1-\mathop{{x}}\nolimits^{{2}}}}}} \text{d} x= \text{arcsin} x+C} \end{aligned} $$ ## 2 三角函数 $$ \begin{aligned} &{{}_{ }^{ } \int _{ }^{ } \text{cos}x \text{d} x= \text{sin} x+C}\\ & {{}_{ }^{ } \int _{ }^{ } \text{sin} x \text{d} x=- \text{cos} x+C}\\ & {{}_{ }^{ } \int _{ }^{ } \text{tan} x \text{d} x=- \text{ln} { \left| { \text{cos} x} \right| }+C}\\ & {{}_{ }^{ } \int _{ }^{ } \text{cot} x \text{d} x= \text{ln} { \left| { \text{sin} x} \right| }+C}\\ & {{}_{ }^{ } \int _{ }^{ } \text{sec} x \text{d} x= \text{ln} { \left| { \text{sec} x+ \text{tan} x} \right| }+C}\\ & {{}_{ }^{ } \int _{ }^{ } \text{csc} x \text{d} x= \text{ln} { \left| { \text{csc} x- \text{cot} x} \right| }+C}\\ & {{}_{ }^{ } \int _{ }^{ }\frac{{1}}{{\mathop{{ \text{cos} }}\nolimits^{{2}}x}} \text{d} x={}_{ }^{ } \int _{ }^{ }\mathop{{ \text{sec} }}\nolimits^{{2}}x \text{d} x= \text{tan} x+C}\\ & {{}_{ }^{ } \int _{ }^{ }\frac{{1}}{{\mathop{{ \text{sin} }}\nolimits^{{2}}x}} \text{d} x={}_{ }^{ } \int _{ }^{ }\mathop{{ \text{csc} }}\nolimits^{{2}}x \text{d} x=- \text{cot} x+C}\\ & {{}_{ }^{ } \int _{ }^{ } \text{sec} x \text{tan} x \text{d} x= \text{sec} x+C}\\ & {{}_{ }^{ } \int _{ }^{ } \text{csc} x \text{cot} x \text{d} x=- \text{csc} x+C}\\ \end{aligned} $$ ## 3 指数函数 $$ \begin{aligned} & {{}_{ }^{ } \int _{ }^{ }\mathop{{e}}\nolimits^{{x}} \text{d} x=\mathop{{e}}\nolimits^{{x}}+C}\\ & {{}_{ }^{ } \int _{ }^{ }\mathop{{a}}\nolimits^{{x}} \text{d} x=\frac{{\mathop{{a}}\nolimits^{{x}}}}{{ \text{ln} a}}+C}\\ & {{}_{ }^{ } \int _{ }^{ } \text{sh} x \text{d} x= \text{ch} x+C}\\ & {{}_{ }^{ } \int _{ }^{ } \text{ch} xdx= \text{sh} x+C} \end{aligned} $$ ## 4 分式 $$ \begin{aligned} & {{}_{ }^{ } \int _{ }^{ }\frac{{1}}{{\mathop{{x}}\nolimits^{{2}}+\mathop{{a}}\nolimits^{{2}}}} \text{d} x=\frac{{1}}{{a}} \text{arctan} \frac{{x}}{{a}}+C}\\ & {{}_{ }^{ } \int _{ }^{ }\frac{{1}}{{\mathop{{x}}\nolimits^{{2}}-\mathop{{a}}\nolimits^{{2}}}} \text{d} x=\frac{{1}}{{2a}} \text{ln} { \left| {\frac{{x-a}}{{x+a}}} \right| }+C}\\ & {{}_{ }^{ } \int _{ }^{ }\frac{{1}}{{\sqrt{{\mathop{{a}}\nolimits^{{2}}-\mathop{{x}}\nolimits^{{2}}}}}} \text{d} x= \text{arcsin} \frac{{x}}{{a}}+C}\\ & {{}_{ }^{ } \int _{ }^{ }\frac{{1}}{{\sqrt{{\mathop{{x}}\nolimits^{{2}}+\mathop{{a}}\nolimits^{{2}}}}}} \text{d} x= \text{ln} { \left( {x+\sqrt{{\mathop{{x}}\nolimits^{{2}}+\mathop{{a}}\nolimits^{{2}}}}} \right) }+C}\\ & {{}_{ }^{ } \int _{ }^{ }\frac{{1}}{{\sqrt{{\mathop{{x}}\nolimits^{{2}}-\mathop{{a}}\nolimits^{{2}}}}}} \text{d} x= \text{ln} { \left( {x+\sqrt{{\mathop{{x}}\nolimits^{{2}}\mathop{{a}}\nolimits^{{2}}}}} \right) }+C} \end{aligned} $$ ## 5 积分性质 $$ \begin{aligned} & {{}_{ }^{ } \int _{ }^{ }{ \left[ {f{ \left( {x} \right) }+g{ \left( {x} \right) }} \right] } \text{d} x={}_{ }^{ } \int _{ }^{ }f{ \left( {x} \right) } \text{d} x+{}_{ }^{ } \int _{ }^{ }g{ \left( {x} \right) } \text{d} x}\\ & {{}_{ }^{ } \int _{ }^{ }kf{ \left( {x} \right) } \text{d} x=k{}_{ }^{ } \int _{ }^{ }f{ \left( {x} \right) } \text{d} x}\\ & {{}_{ }^{ } \int _{ }^{ }u \text{d} v=uv-{}_{ }^{ } \int _{ }^{ }v \text{d} u} \end{aligned} $$ ## 换元法