Why is it that integration of 1/(xlog x) is log(log x) and not log(log x) (the integration of log x)?

hrostent72t

hrostent72t

Answered question

2022-12-20

Why is it that integration of 1 x log x is log ( log x ) and not log ( log x ) (the integration of log x)?

Answer & Explanation

Camryn Moreno

Camryn Moreno

Beginner2022-12-21Added 4 answers

Let u = log x. Then d u = 1 x d x. We need to determine du in order to take into account (reverse, so to speak) the use of the chain rule involved in differentiating the desired function.
Back to the integral: By substitution, we get
1 x log x d x = 1 log x 1 x d x = 1 u d u
This, in turn is equal to log | u | + C = log | ( log x ) | + C
Indeed, if we differentiate:
( log | log ( x ) | + C ) = 1 log x ( log x ) = 1 log x 1 x = 1 x log x

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