Antiderivative of log(x) without PartsI 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.

Question

Zackery Harvey · Accepted Answer

Step 1This may not necessarily be what you&#039;re looking for, but I got a kick out of it so I figured I&#039;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 2Integration 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 · Accepted Answer

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