含2kπ-α诱导类型三角函数的不定积分

1、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、图例解析如下：

1、∫sin(2kπ-α)dα

=-∫sin(2kπ-α)d(2kπ-α)

=cos(2kπ-α)+c

=cosα+c

2、图例解析如下：

1、∫cos(2kπ-α)dα

=-∫cos(2kπ-α)d(2kπ-α)

=-sin(2kπ-α)+c

=sinα+c

2、图例解析如下：

1、∫tan(2kπ-α)dα

=-∫tan(2kπ-α)d(2kπ-α)

=-∫[sin(2kπ-α) d(2kπ-α)/ cos(2kπ-α)]

=∫d cos(2kπ-α)/cos(2kπ-α)

=ln|cos(2kπ-α)|+c

=ln|cosα|+c

2、图例解析如下：

1、∫cot(2kπ-α)dα

=-∫cot(2kπ-α)d(2kπ-α)

=-∫[cos(2kπ-α) d(2kπ-α)/ sin(2kπ-α)]

=-∫d sin(2kπ-α)/sin(2kπ-α)

=-ln|sin(2kπ-α)|+c

=-ln|sinα|+c

2、图例解析如下：

1、∫sec(2kπ-α)dα

=-∫d(2kπ-α)/ cos(2kπ-α)

=-∫cos(2kπ-α)d(2kπ-α)/ [cos(2kπ-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：

1、∫csc(2kπ-α)dα

=-∫csc(2kπ-α)d(2kπ-α)

=-∫d(2kπ-α)/ sin(2kπ-α)

=-∫sin(2kπ-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：