How to integrate ln x 2 x ?

Question

How to integrate                               ln          x                            2        x            ?

Brian Blackburn · Accepted Answer

Knowing that if             f              (        x        )              =          ln      x       then       f    ´          (      x      )        =          1      x       and with the chain ruleif             f              (        x        )              =          ln              (                  g                      (            x            )                          )             then       f    ´          (      x      )        =                  g        ´                  (          x          )                            g                  (          x          )                    Then       ∫                  ln        x                    2        x                    d      x        =          1      4                      ln        2            x        +    C

jdkhoo88o8j · Accepted Answer

Let,       I    =    ∫                  ln        x                    2        x                    d      x        =          1      2        ∫          ln      x        ⋅          1      x              d      x      . Now, we use IBP :       ∫    u    ⋅    v    ′          d      x        =    u    ⋅    v    -    ∫    u    ′    ⋅    v          d      x      , with,      u    =          ln      x                    and              v    ′    =          1      x      .      ⇒    u    ′    =          1      x                    and              v    =    ∫          1      x              d      x        =          ln      x      .Hence,       I    =          1      2              [              (                  ln          x                )                    (                  ln          x                )            -      ∫              (                  1          x                ⋅                  ln          x                )                    d        x            ]        ,    i    .    e    .    ,        I    =          1      2                      (                  ln          x                )            2        -          1      2        ∫          ln      x        ⋅          1      x              d      x        ,                or              ,        I    =          1      2                      (                  ln          x                )            2        -    I  .      ∴    2    I    =          1      2                      (                  ln          x                )            2      .      ⇒    I    =          1      4                      (                  ln          x                )            2        +    C  .

How to integrate (lnx)/(2x)?

Answered question

Answer & Explanation

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