Find the antiderivatives ∫ ( 2 x 2 + 3 ) 1 / 3 x d xI have hit this in my book and the way I do it I get 3 / 4 ( 2 x 2 + 3 ) 4 / 3 x 2 + cBut my book tells me it should be 3 / 16 ( 2 x 2 + 3 ) 4 / 3

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Find the antiderivatives  ∫              (      2              x        2            +      3      )              1              /            3        x    d  xI have hit this in my book and the way I do it I get  3      /    4            (      2              x        2            +      3      )              4              /            3            x    2    +  cBut my book tells me it should be   3      /    16  (  2      x    2    +  3      )          4              /            3

Raven Higgins · Accepted Answer

Step 1Make the substitution   u  =      x    2    +  3. Then   d  u  =  4  x    d  x so   x    d  x  =      1    4      d  u. Thus   ∫  (  2      x    2    +  3      )          1              /            3        x    d  x  =  ∫      1    4        u          1              /            3          d  u  .Step 2Remark: The derivative of   2      x    2    +  3 is &quot;almost&quot; part of your expression. sure, we have an x instead of 4x, but that&#039;s no problem, since   x  =      1    4    (  4  x  ). In this kind of situation, substitution is often useful. You will meet variants often, like   ∫  x      e                  x        2                d  x

lobht98 · Accepted Answer

Explanation:Use the substitution   u  =  2      x    2    +  3  ,  d  u  =  4  x. It is unfortunately not true that   ∫  f  (  x  )  g  (  x  )  d  x  =  ∫  f  (  x  )  d  x  ∫  g  (  x  )  d  x; you cannot integrate the x part separately.

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