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