f(sinx)^2 导数=(cosx)^2+(tanx)^2 f(X)=?

问题描述:

f(sinx)^2 导数=(cosx)^2+(tanx)^2 f(X)=?

∵f′[(sinx)^2]=(cosx)^2+(tanx)^2=1-(sinx)^2+(sinx)^2/[1-(sinx)^2],
∴f′(x)=1-x+x/(1-x)=1-x-x/(x-1)=1-x-(x-1+1)/(x-1)=-x-1/(x-1).
∴f(x)=-∫xdx-∫[1/(x-1)]dx=-(1/2)x^2-ln|x-1|+C,其中C为任意常数.