int (ln^3 x)/(x) dx Is this the same as int ((ln x)^3)/(x)dx?

scherezade29pc 2022-07-21 Answered
Help me with this
ln 3 x x   d x
Is this the same as
( ln x ) 3 x   d x ?
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Answers (1)

akademiks1989rz
Answered 2022-07-22 Author has 16 answers
Unfortunately, "shorthand" can lead to ambiguity:
ln 3 x = ( ln ( ln ( ln x ) ) ? or ln 3 x = ( ln x ) 3 ?
But as you suspected, in this context, and given the integral, I'm am quite sure that ln 3 x = ( ln x ) 3 , much like sin 2 ( x ) = ( sin x ) 2 . So your integral amounts to:
ln 3 x x   d x = ( ln x ) 3 x d x
Let u = ln ( x ) , ; d u = d x x
( ln x ) 3 x d x = u 3 d u = 1 4 u 4 + C
= 1 4 ( ln x ) 4 + C = 1 4 ln 4 x + C
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