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
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)
=sina(2kπ+α)+c
=sinα+c
2、图例解析如下:
1、 ∫tan(2kπ+α)dα
=∫[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、∫ctg(2kπ+α)dα
=∫[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
=∫dα/ cos(2kπ+α)
=∫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+sinα)/(1-sinα)]+c
=(1/2)ln[(1+sinα)^2/(cosα)^2]+c
=ln|(1+sinα)/cosα|+c
=ln|secα+tana|+c
2、图例解析如下:
1、∫csc(2kπ+α)dα
=∫dα/ sin(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+cosα)/(1-cosα)]+c
=-(1/2)ln[(1+cosα)^2/(sinα)^2]+c
=-ln|(1+cosα)/sinα|+c
=-ln|cscα+ctga|+c
2、图例解析如下: