What is the antiderivative of ln ( x ) 2 ?

Question

What is the antiderivative of                               ln                      (            x            )                          2                  ?

Shiloh Hinton · Accepted Answer

Here,      I    =    ∫                            ln                      (            x            )                          2                    d        x            Using Power-coefficient Rule :                                                        ln              x                        n                    =                n        ⋅                  ln          x                     , we get      I    =    ∫                            ln          x                2                    d        x              =    ∫    2    ⋅          ln      x        ...    →          (      A      )      Using , Integration by parts:                    ∫                  (          u          ⋅          v          )                          d          x                =        u        ∫        v                  d          x                -        ∫                  (          u          ′          ∫          v                      d            x                    )                          d          x                          u    =          ln      x                    and              v    =    2    ⇒    u    ′    =          1      x                    and              ∫    v          d      x        =    2    x        I    =          ln      x        ⋅    2    x    -    ∫          1      x        ⋅    2    x          d      x        +    c        ⇒    I    =    2    x    ⋅          ln      x        -    ∫    2          d      x        +    c        ⇒    I    =    2    x          ln      x        -    2    x    +    c        ⇒    I    =    2    x          [              ln        x            -      1      ]        +    c  ...........................................................................................................Note:If       I    =                  (                  ln          x                )            2       ,then      I    =                  (                  ln          x                )            2        ⋅    1          d      x      Using , Integration by parts:      u    =                  (                  ln          x                )            2                    and              v    =    1    ⇒    u    ′    =                  2                  ln          x                    x                    and              ∫    v          d      x        =    x        I    =                  (                  ln          x                )            2        ∫    1          d      x        -    ∫                  2                  ln          x                    x        ⋅    x          d      x            =                  (                  ln          x                )            2        ⋅    x    -    ∫    2    ⋅          ln      x      Using the aforementioned outcome for             (      A      )             I    =    x    ⋅                  (                  ln          x                )            2        -          {      2      x              [                  ln          x                -        1        ]            }        +    c

Xander Torres · Accepted Answer

A function&#039;s antiderivative is essentially the integral of the function. So we get:      ∫                            ln                      (            x            )                          2                           d      x      I&#039;m assuming that we have       ∫                            ln          x                2                           d      x      .Using logarithm rules, we get:      =    ∫    2          ln      x                   d      x            =    2    ∫          ln      x                   d      x      This is a common integral, where       ∫          ln      x                   d      x        =    x          ln      x        -    x    +    C  .      ∴    =    2    x          ln      x        -    2    x    +    C

What is the antiderivative of ln(x)^2?

Answered question

Answer & Explanation

New Questions in Calculus 1