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Assume that ... and that ... . The integral of ... is ... , where ... is an arbitrary constant. The integral of ... is ... , where again ... is an arbitrary constant and ... is the natural logarithm of ... , often written as ... . When ... is close to zero, ... and ... are close, so there must be some connection between their integrals! Choose ... and ... so that the two integrals are both zero at ... . The integrals are then ... and ... . For ... close to zero these functions are very close; in symbols, ... . Using the difference quotient for the derivative of the base- ... exponential function ... with respect to ... (not ... ) and using ... instead of the more usual ... gives ... . This is more usually written with ... as the variable: ... , with the special case ...

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      EUN,LOM,LRE4,work-cmr-id:398051,http://demonstrations.wolfram.com:http://demonstrations.wolfram.com/TheNaturalLogarithmIsTheLimitOfTheIntegralsOfPowers/,ilox,learning resource exchange,LRE metadata application profile,LRE

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