Using the FTOC to find the derivative of the integral ∫ x 2 5 ( 4 x + 2 ) d x .

Question

Using the FTOC to find the derivative of the integral      ∫                  x        2              5    (  4  x  +  2  )    d  x  .

Patricia Curry · Accepted Answer

You have  f  (  x  )  =      ∫                  x        2              5    4  t  +  2    d  tand you want to find       f    ′    (  x  ).First note that  f  (  x  )  =  −      ∫    5          g      (      x      )        4  t  +  2    d  twhere   g  (  x  )  =      x    2  . So   f  (  x  ) is the composition of two functions:  f  (  x  )  =  h  (  g  (  x  )  )  .where                    h        (        x        )                            =        −                  ∫          5          x                4        x        +        2                d        t                and                            g        (        x        )                            =                  x          2                .            So by the chain rule you have      f    ′    (  x  )  =      h    ′    (  g  (  x  )  )            g      ′        (    x    )    =  −  (  4  g  (  x  )  +  2  )            g      ′        (    x    )    =  …

Using the FTOC to find the derivative of the integral <msubsup> ∫

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