Question

高等数学利用x=arctant带入方程，想问问cosarctant和sinarctant怎么个化法

Answer 1

令y=arctane^x,则e^x=tany,x=ln(tany) dx=cotysec^2ydy 原式=∫ycot^2y*cotysec^2ydy =∫ycsc^2ycotydy =∫ycosy/sin^3ydy =∫y/sin^3ydsiny =(-1/2)∫yd(1/sin^2y) =(-1/2)y/sin^2y+1/2∫dy/sin^2y =(-1/2)ycsc^2y-1/2coty+C =(-1/2)arctane^xcsc^2(arctane^x)-1/2cot(arctane^x)+C =(-1/2)arctane^x[1+e^(-2x)]-(1/2)e^(-x)+C =(-1/2)[e^-(2x)*arctane^x+arctane^x+e^(-x)]+C 或者令e^x=t x=lnt ∫arctane^x/e^2xdx=∫arctant/t^3dt=-1/2∫arctantd(1/t^2) =-1/2[(arctant/t^2)-∫1/(t^2)(1+t^2)dt] =-1/2{(arctant/t^2)-∫[1/(t^2)]-[1/(1+t^2)]dt} =-1/2[(arctant/t^2)+1/t+arctant+c] =-1/2[(arctane^x/e^2x)+1/e^x+arctane^x]+c

追问

不用长篇大论了，画图就搞定了