求定积分.(e^arctan x)/[(x^2+1)^(3/2)]的原函数

___岁月_如歌_ 2023-03-18 13:29

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  _M30_

  2023-03-18 13:35

  ∵∫e^y*cosydy=e^y*(cosy+siny)/2+C (C是积分常数)

  ∴ ∫e^(arctanx)dx/(x²+1)^(3/2)

  =∫e^y*sec²ydy/sec³y (令y=arctanx,则cosy=1/√(x²+1),siny=x/√(x²+1))

  =∫e^y*cosydy

  =e^y*(cosy+siny)/2+C (C是积分常数)

  =e^(arctanx)[1/√(x²+1)+x/√(x²+1)]/2+C

  =(x+1)e^(arctanx)/[2√(x²+1)]+C.
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