Antiderivative of log(x) without PartsI understand how the antiderivative of log(x) can be obtained by...

Joshua Foley

Joshua Foley

Answered

2022-07-01

Antiderivative of log(x) without Parts
I understand how the antiderivative of log(x) can be obtained by Integration by Parts (i.e. product rule), but I was wondering how-if at all- it could be obtained only using sum/difference rule and substitution/chain rule.

Answer & Explanation

Zackery Harvey

Zackery Harvey

Expert

2022-07-02Added 21 answers

Step 1
This may not necessarily be what you're looking for, but I got a kick out of it so I figured I'd share. We may evaluate the antiderivative by using integration by parts indirectly.
We seek the integral I f ( x ) = f 1 ( x ) d x ..
We set x = f ( u ) to get I f ( x ) = u f ( u ) d u ..
Step 2
Integration by parts gives I f ( x ) = u f ( u ) f ( u ) d u which is I f ( x ) = x f 1 ( x ) F f 1 ( x ) + C ..
Choosing f 1 ( x ) = ln x ,, I ln ( x ) = x ln x x + C ..
Janet Forbes

Janet Forbes

Expert

2022-07-03Added 4 answers

Explanation:
1 t ln ( x ) d x = 1 t 1 x 1 u d u d x = 1 t u t 1 u d x d u = 1 t t u u d u = [ t ln ( u ) u ] 1 t

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