本经验介绍含2kπ-α诱导类型三角函数的不定积分，即求∫sin(2kπ-α)dα，∫cos(2kπ-α)dα，∫tan（2kπ-α)dα，∫cot(2kπ-α）dα，∫sec(2kπ-α)dα，∫csc(2kπ-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2kπ-α三角函数的诱导公式
1、sin(2kπ-α)=-sin αcos(2kπ-α)=cos αtan(2kπ-α)=-tan αcot(2kπ-α)=-cot αsec(2kπ-α)=sec αcsc(2kπ-α)=-csc α
2、图例解析如下：
2.sin(2kπ-α)的不定积分
1、∫sin(2kπ-α)dα=-∫sin(2kπ-α)d(2kπ-α)=cos(2kπ-α)+c=cosα+c
2、图例解析如下：
3.cos(2kπ-α)的不定积分
1、∫cos(2kπ-α)dα=-∫cos(2kπ-α)d(2kπ-α)=-sin(2kπ-α)+c=sinα+c
2、图例解析如下：
4.tan(2kπ-α)的不定积分
1、∫tan(2kπ-α)d拿骛蟊痊α=-∫tan(2kπ-α)d(2kπ-α)=-∫[sin(2kπ-α)d(2kπ-α)/ cos(2kπ-珍提疮翘α)]=∫d cos(2kπ-α)/cos(2kπ-α)=ln|cos(2kπ-α)|+c=ln|cosα|+c
2、图例解析如下：
5.cot(2kπ-α)的不定积分
1、∫cot(2kπ-α)d拿骛蟊痊α=-∫cot(2kπ-α)d(2kπ-α)=-∫[cos(2kπ-α)d(2kπ-α)/ sin(2kπ-珍提疮翘α)]=-∫d sin(2kπ-α)/sin(2kπ-α)=-ln|sin(2kπ-α)|+c=-ln|sinα|+c
2、图例解析如下：
6.sec(2kπ-α)的不定积分
1、∫sec(2kπ-α)d拿骛蟊痊α=-∫d(2kπ-α)/ cos(2kπ-α)=-∫cos(2kπ-α)d(2kπ-α)/ [cos(2k嗝搜肠怵π-α)]^2=-∫dsin(2kπ-α)/ [1-(sin(2kπ-α))^2}=-∫dsin(2kπ-α)/ [(1-sin(2kπ-α))(1+ sin(2kπ-α))]=-(1/2)[∫dsin(2kπ-α)/ (1-sin(2kπ-α))+∫dsin(2kπ-α)/ (1+sin(2kπ-α))]=-(1/2)ln{[1+sin(2kπ-α)]/ [1-sin(2kπ-α)]}+c=-(1/2)ln[(1+sin(2kπ-α))/(1-sin(2kπ-α))]+c=-(1/2)ln[(1+sin(2kπ-α))^2/(cos(2kπ-α))^2]+c=-ln|(1+sin(2kπ-α))/cos(2kπ-α)|+c=-ln|(1-sinα)/cosα|+c=-ln|secα-tanα|+c
2、图例解析如下：
7.csc(2kπ-α)的不定积分
1、∫csc(2kπ-α)d拿骛蟊痊α=-∫csc(2kπ-α)d(2kπ-α)=-∫d(2kπ-α)/ sin(2kπ-α)=-∫sin(2k嗝搜肠怵π-α)d(2kπ-α)/ [sin(2kπ-α)]^2=∫dcos(2kπ-α)/ [1-(cos(2kπ-α))^2]=∫dcos(2kπ-α)/ [(1-cos(2kπ-α))(1+ cos(2kπ-α))]=(1/2)[∫dcos(2kπ-α)/ (1-cos(2kπ-α))+∫dcos(2kπ-α)/ (1+cos(2kπ-α))]=(1/2)ln[(1+cos(2kπ-α))/ (1-cos(2kπ-α))]+c=(1/2)ln[(1+cos(2kπ-α))^2/(sin(2kπ-α))^2]+c=ln|(1+cos(2kπ-α))/sin(2kπ-α)|+c=ln|(1+cosα)/sinα|+c=ln|cscα+cota|+c
2、图例解析如下：
