含-α诱导类型三角函数的不定积分

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：

3.cos(-α)dα

1、∫cos(-α)dα=-∫cos(-α)d(-α)=-sin(-α)+c=sinα+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：

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：