Antiderivative of ( x 2 + c ) − 3 / 2 What method should be used to determine the antiderivative of this expression?Edit: I have c &gt; 0 in the problem I'm working on.

Question

Antiderivative of   (      x    2    +  c      )          −      3              /            2      What method should be used to determine the antiderivative of this expression?Edit: I have   c  &amp;gt;  0 in the problem I&#039;m working on.

Duncan Cox · Accepted Answer

Step 1  ∫  (      x    2    +  c      )          −      3              /            2        d  x  =  ∫      1          (              x        2            +      c              )        3            Let us use a new variable t such that             x      2        +    c    =  x  +  t. This is the first Euler substitution.From this formula we can solve for x to get   x  =            c      −              t        2                    2      t      ,   d  x  =            −      c      −              t        2                    2              t        2              d  t,             x      2        +    c    =  x  +  t  =            c      +              t        2                    2      t      , and       (          x      2        +    c          )      3        =            (      c      +              t        2                    )        3                    8              t        3            .Step 2We put these into the integral and get  ∫            8              t        3                    (      c      +              t        2                    )        3                        −      (      c      +              t        2            )              2              t        2              d  t  =  −  2  ∫            2      t              (      c      +              t        2                    )        2              d  t  =  2      1          c      +              t        2            We can return to the variable x since   t  =            x      2        +    c    −  x.Therefore,   ∫  (      x    2    +  c      )          −      3              /            2        =      2          c      +                        (                                    x              2                        +            c                    −          x          )                2

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