Question

高等数学不定积分的计算？

请问这个题怎么做？请写下详细步骤，谢谢！

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

Answer 2

可令e^x＝t，x＝lnt，代入原式分部积分即可。