Question

高数不定积分

高数不定积分求解

Answer 1

差不多

Answer 2

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

厉害了

Answer 3

先拆分

追答