arctane等于什么 arctane等于多少pai

arctane等于多少 arctane=1.2182829(弧度)=69.8025(角度)

arctane^x+arctane^(-x)=π/2.公式推导:设arctanA=a,arctan(1/A)=b.则tana=A,tanb=1/A,即tana*tanb=1.那么:sinasinb=cosacosb.所以cos(a+b)=0,a+b=π/2.既然有.

设arctanx=A,arctan1/x=B 则tanA=x,tanB=1/x tanA*tanB=1(sinA*sinB)/(cosA*cosB)=1(cosA*cosB)-(sinA*sinB)=0 cos(A+B)=0 A+B=π/2

令f(x)=arctane^x + arctane^-x f'(x)=1/(1+e^2x) *e^x +1/(1+e^-2x) *[-e^(-x)]=e^x /(1+e^2x) -e^(-x)/(1+e^-2x) =e^x /(1+e^2x) -e^(x)/(1+e^2x)=0 所以 f(x)=常数 令x=0 f(0)=arctan1+arctan1=π/4+π/4=π/2 从而 arctane^x + arctane^-x≡π/2

y = arctane^(-x), y' = {1/[1+e^(-2x)]}*[e^(-x)]*(-1) = -1/[(e^x)+e^(-x)]

arctanex*2-x =arctanex+arctane-x =arctanex-x+arctane =arctane+arctane =2arctane

等于π/2

令u=e^x,则e^(-x)=1/u,dx=du/u,x∈[0,1]时,u∈[1,e] ∫(0→1) dx/[e^x+e^(-x)] =∫(1→e) du/[u+(1/u)]u =∫(1→e) du/(u²+1) =arctanu|(1→e) =arctane-arctan1 =arctane-π/4

z=(x^2+y^2)e^[-arctan(y/x)] ?z/?x=(2x)e^[-arctan(y/x)]-(x^2+y^2)e^[-arctan(y/x)](-y/x^2)/[1+y^2/x^2] =(2x)e^[-arctan(y/x)]+ye^[-arctan(y/x)]=(2x+y)e^[-arctan(y/x)] ?z/?y=(2y)e^[-arctan(y/x)]-(x^2+y^2)e^[-arctan(y/x)](1/x)/[1+y^2/x^2] =(2y)e^[-arctan(y/x)]-xe^[-arctan(y/x)]=(2y-x)e^[-arctan(y/x)]

∫(arctane^x/e^x)dx=-1/2∫arctane^xde^(-2x) =-1/2{e^(-2x)arctane^x-∫de^x/[e^2x(1+e^2x)]}=-1/2e^(-2x)arctane^x+1/2∫(1/e^2x-1/(1+e^2x))dx=-1/2e^(-2x)arctane^x-1/(2e^x)-1/2.