Why can't you integrate all power functions without a log function? You need a logarithm functio

m4tx45w

m4tx45w

Answered question

2022-01-23

Why cant

Answer & Explanation

Emilie Booker

Emilie Booker

Beginner2022-01-24Added 14 answers

The problem is this: you know that for n1, the indefinite integral of xn is equal to xn+1n+1. But this formula can't possibly be valid for n=1 because the denominator vanishes. So instead you have to take the limit. It's easiest to see how this works with the definite integral
abxndx=an+1bn+1n+1.
If you want to see what happens at n=1, what you do is to take the limit as n1. By l'Hopital's rule, remembering that xk=eklnx, we find that
limn1an+1bn+1n+1=limn1an+1lnabn+1lnb1=lnalnb.So that's where the logarithm appears: it naturally comes out of the value of this limit, and in fact this limit can be used to define the logarithm.
More generally speaking, if you have a collection of functions closed under differentiation, you are in no way guaranteed that that collection of functions is also closed under integration. In fact given a class of functions, integration generally gives you new functions not in that class.

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