Antiderivative definitionRecently, I was trying to derive the antiderivative of 1 x by taking lim p → − 1 ∫ x p but quickly realized that it would not work as it gives x 0 0 however, if I instead took lim p → − 1 ∫ a x x p .This gives lim p → − 1 x p + 1 − a p + 1 p + 1 Which evaluated with L'Hopital's rule returned ln ⁡ ( x ) − ln ⁡ ( a ) where ln(a) is just some constant and returns the expected result.This leads me to ask, is the antiderivative defined as a definite integral with variable upper bound, or is there something else I'm not understanding here? I looked at the fundamental theorem of calculus, for hints and this seems to be the case, but there was nothing definitive that said this was true.

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Antiderivative definitionRecently, I was trying to derive the antiderivative of       1    x   by taking       lim          p      →      −      1            ∫            x      p       but quickly realized that it would not work as it gives             x      0        0   however, if I instead took       lim          p      →      −      1                ∫    a    x              x      p      .This gives       lim          p      →      −      1                              x                  p          +          1                    −              a                  p          +          1                            p      +      1      Which evaluated with L&#039;Hopital&#039;s rule returned   ln  ⁡  (  x  )  −  ln  ⁡  (  a  ) where ln(a) is just some constant and returns the expected result.This leads me to ask, is the antiderivative defined as a definite integral with variable upper bound, or is there something else I&#039;m not understanding here? I looked at the fundamental theorem of calculus, for hints and this seems to be the case, but there was nothing definitive that said this was true.

Nola Rivera · Accepted Answer

Step 1  ∫      1    x    d  x  =  ln  ⁡      |    x      |    +  C if this is true, then:      1    x    =      d          d      x            [    ln    ⁡          |        x          |        ]  Step 2Let   y  =  ln  ⁡  (  x  )then       e    y    =  x      e    y              d      y              d      x        =  1            d      y              d      x        =      1          e      y        =      1    x

Antiderivative definition Recently, I was trying to derive the antiderivative of 1 x

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