本经验介绍含2π+α诱导类型三角函数的不定积分，即求∫sin(2息遴颞阈π+α)dα，∫cos(2π+α)dα，∫tan(2π+α)dα，∫cot(2π+α)dα，∫s髫潋啜缅ec(2π+α)dα，∫csc(2π+α)dα的步骤。
工具/原料
三角函数基本知识
不定积分基本知识
1.含2π+α的诱导公式
1、sin(2π+α)=sin αcos(2π+α)=cos αtan(2π+α)=tan αcot(2π+α)=cot αsec(2π+α)=sec αcsc(2π+α)=csc α
2、图例解析如下：
2.sin(2π+α)的不定积分
1、∫sin(2π+α)dα=∫sin(2π+α)d(2π+α)=-cos(2π+α)+c=-cosα+c
2、图例解析如下：
3.cos(2π+α)dα
1、∫cos(2π+α)dα=∫cos(2π+α)d(2π+α)=sin(2π+α)+c=sinα+c
2、图例解析如下：
4.tan(2π+α)dα
1、∫tan(2π+α)dα屏顿幂垂=∫[sin(2π+α)d(2π+α)/ cos(2π+α)]=-∫d cos(2π+α)/cos(2π+α)=-ln|cos(2π+α)|+c=-ln|cosα|+c
2、图例解析如下：
5.cot(2π+α)dα
1、∫cot(2π+α)dα=∫[cos(2π+α)d(2π+α)/ sin(2π+α)]=∫d sin(2π+α)/sin(2π+α)=ln|sin(2π+α)|+c=ln|sinα|+c
2、图例解析如下：
6.sec(2π+α)dα
1、∫sec(2π+α)dα屏顿幂垂=∫dα/ cos(2π+α)=∫d(2π+α)/ cos(2π+α)=∫cos(2π+α)d(2π敫苻匈酃+α)/ [cos(2π+α)]^2=∫dsin(2π+α)/ {1-[sin(2π+α)]^2}=∫dsin(2π+α)/ {[1-sin(2π+α)][1+ sin(2π+α)]}=(1/2){∫dsin(2π+α)/ [1-sin(2π+α)]+∫dsin(2π+α)/ [1+sin(2π+α)]}=(1/2)ln{[1+sin(2π+α)]/ [1-sin(2π+α)]}+c=(1/2)ln[(1+sinα)/(1-sinα)]+c=(1/2)ln[(1+sinα)^2/(cosα)^2]+c=ln|(1+sinα)/cosα|+c=ln|secα+cotα|+c
2、图例解析如下：
7.csc(2π+α)dα
1、∫csc(2π+α)dα屏顿幂垂=∫dα/ sin(2π+α)=∫d(2π+α)/ sin(2π+α)=∫sin(2π+α)d(2π敫苻匈酃+α)/ [sin(2π+α)]^2=-∫dcos(2π+α)/ {1-[cos(2π+α)]^2}=-∫dcos(2π+α)/ {[1-cos(2π+α)][1+ cos(2π+α)]}=-(1/2){∫dcos(2π+α)/ [1-cos(2π+α)]+∫dcos(2π+α)/ [1+cos(2π+α)]}=-(1/2)ln{[1+cos(2π+α)]/ [1-cos(2π+α)]}+c=-(1/2)ln[(1+cosα)/(1-cosα)]+c=-(1/2)ln[(1+cosα)^2/(sinα)^2]+c=-ln|(1+cosα)/sinα|+c=-ln|cscα+cota|+c
2、图例解析如下：
