含2π-α诱导类型三角函数的不定积分

1、sin(2π-α)=-sin α

cos(2π-α)=cos α

tan(2π-α)=-tan α

cot(2π-α)=-cot α

sec(2π-α)=sec α

csc(2π-α)=-csc α

2、图例解析如下：

1、∫sin(2π-α)dα

=-∫sin(2π-α)d(2π-α)

=cos(2π-α)+c

=cosα+c

2、图例解析如下：

1、 

∫cos(2π-α)dα

=-∫cos(2π-α)d(2π-α)

=-sin(2π-α)+c

=sinα+c

2、图例解析如下：

1、∫tan(2π-α)dα

=-∫tan(2π-α)d(2π-α)

=-∫[sin(2π-α) d(2π-α)/ cos(2π-α)]

=∫d cos(2π-α)/cos(2π-α)

=ln|cos(2π-α)|+c

=ln|cosα|+c

2、图例解析如下：

1、∫cot(2π-α)dα

=-∫cot(2π-α)d(2π-α)

=-∫[cos(2π-α) d(2π-α)/ sin(2π-α)]

=-∫d sin(2π-α)/sin(2π-α)

=-ln|sin(2π-α)|+c

=-ln|sinα|+c

2、图例解析如下：

1、∫sec(2π-α)dα

=-∫d(2π-α)/ cos(2π-α)

=-∫cos(2π-α)d(2π-α)/ [cos(2π-α)]^2

=-∫dsin(2π-α)/ [1-(sin(2π-α))^2}

=-∫dsin(2π-α)/ [(1-sin(2π-α))(1+ sin(2π-α))]

=-(1/2)[∫dsin(2π-α)/ (1-sin(2π-α))+∫dsin(2π-α)/ (1+sin(2π-α))]

=-(1/2)ln{[1+sin(2π-α)]/ [1-sin(2π-α)]}+c

=-(1/2)ln[(1+sin(2π-α))/(1-sin(2π-α))]+c

=-(1/2)ln[(1+sin(2π-α))^2/(cos(2π-α))^2]+c

=-ln|(1+sin(2π-α))/cos(2π-α)|+c

=-ln|(1-sinα)/cosα|+c

=-ln|secα-tanα|+c

2、图例解析如下：

1、∫csc(2π-α)dα

=-∫csc(2π-α)d(2π-α)

=-∫d(2π-α)/ sin(2π-α)

=-∫sin(2π-α)d(2π-α)/ [sin(2π-α)]^2

=∫dcos(2π-α)/ [1-(cos(2π-α))^2]

=∫dcos(2π-α)/ [(1-cos(2π-α))(1+ cos(2π-α))]

=(1/2)[∫dcos(2π-α)/ (1-cos(2π-α))+∫dcos(2π-α)/ (1+cos(2π-α))]

=(1/2)ln[(1+cos(2π-α))/ (1-cos(2π-α))]+c

=(1/2)ln[(1+cos(2π-α))^2/(sin(2π-α))^2]+c

=ln|(1+cos(2π-α))/sin(2π-α)|+c

=ln|(1+cosα)/sinα|+c

=ln|cscα+cota|+c

2、图例解析如下：