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Integrate  dx  / √(1 + x²)

let   x = tan y
      dx = sec² y  dy

f(x) dx  =  sec² y  dy / sec y  = sec y  dy
The formula for the Integral of secy dy
  = Log | Sec y + Tan y |  + K
  = Log | √(1+x²)  + x  |  + K

You can differentiate it once and check the answer.
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