cosx分之一的导数是:
y=1/cosx=(cosx)^(-1)
所以y=-1*(cosx)^(-2)*(cosx)
=-1/cosx*(-sinx)
=(sinx/cosx)(1/cosx)
=tanxsecx
1/cosx的导数是cosx分之一:
∫1/cosx dx=∫cosx/(1-sin²x)
dx=(1/2)∫[1/(1-sinx)+1/(1+sinx)]
dsinx=(1/2)ln[(1+sinx)/(1-sinx)]+C
所以(1/2)ln[(1+sinx)/(1-sinx)]+C的导数是1/cosx
1/cosx的原函数是ln|secx+tanx|+C。解答如下:
先算1/sinx原函数,S表示积分号
S1/sinxdx
=S1/(2sin(x/2)cos(x/2))dx
=S1/[tan(x/2)cos²(x/2)]d(x/2)
=S1/[tan(x/2)]d(tan(x/2))
=ln|zhitan(x/2)|+C
因为tan(x/2)=sin(x/2)/cos(x/2)=2sin²(x/2)/[2sin(x/2)cos(x/2)]=(1-cosx0/sinx=cscx-cotx
所以S1/sinxdx=ln|cscx-cotx|+C
S1/cosxdx
=S1/sin(x+派/2)d(x+派/2)
=ln|csc(x+派/2)-cot(x+派/2)|+C
=ln|secx+tanx|+C