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more三角函数基本知识不定积分基本知识1.含2kπ+α的诱导公式1/1分步阅读

sin(2kπ+α)=sin α

cos(2kπ+α)=cos α

tan(2kπ+α)=tan α

cot(2kπ+α)=cot α

sec(2kπ+α)=sec α

csc(2kπ+α)=csc α

2.sin(2kπ+α)的不定积分1/2

∫sin(2kπ+α)dα

=∫sin(2kπ+α)d(2kπ+α)

=-cos(2kπ+α)+c

=-cosα+c

2/2

图例解析如下：

3.cos(2kπ+α)的不定积分1/2

∫cos(2kπ+α)dα

=∫cos(2kπ+α)d(2kπ+α)

=sina(2kπ+α)+c

=sinα+c

2/2

图例解析如下：

4.tan(2kπ+α)的不定积分1/2

∫tan(2kπ+α)dα

=∫

=-∫d cos(2kπ+α)/cos(2kπ+α)

=-ln|cos(2kπ+α)|+c

=-ln|cosα|+c

2/2

图例解析如下：

5.cot(2kπ+α)的不定积分1/2

∫ctg(2kπ+α)dα

=∫

=∫d sin(2kπ+α)/sin(2kπ+α)

=ln|sin(2kπ+α)|+c

=ln|sinα|+c

2/2

图例解析如下：

6.sec(2kπ+α)的不定积分1/2

∫sec(2kπ+α)dα

=∫dα/ cos(2kπ+α)

=∫d(2kπ+α)/ cos(2kπ+α)

=∫cos(2kπ+α)d(2kπ+α)/ ^2

=∫dsin(2kπ+α)/ {1-^2}

=∫dsin(2kπ+α)/ {}

=(1/2){∫dsin(2kπ+α)/ +∫dsin(2kπ+α)/ }

=(1/2)ln{/ }+c

=(1/2)ln+c

=(1/2)ln+c

=ln|(1+sinα)/cosα|+c

=ln|secα+tana|+c

2/2

图例解析如下：

7.csc(2kπ+α)的不定积分1/2

∫csc(2kπ+α)dα

=∫dα/ sin(2kπ+α)

=∫d(2kπ+α)/ sin(2kπ+α)

=∫sin(2kπ+α)d(2kπ+α)/ ^2

=-∫dcos(2kπ+α)/ {1-^2}

=-∫dcos(2kπ+α)/ {}

=-(1/2){∫dcos(2kπ+α)/ +∫dcos(2kπ+α)/ }

=-(1/2)ln{/ }+c

=-(1/2)ln+c

=-(1/2)ln+c

=-ln|(1+cosα)/sinα|+c

=-ln|cscα+ctga|+c

2/2

图例解析如下：

函数三角函数诱导公式不定积分