Find the derivative of the following function using the Fundamental Theorem of Calculus: F ( x ) = ∫ x 3 x 6 ( 2 t − 1 ) 3 d t

Question

Find the derivative of the following function using the Fundamental Theorem of Calculus:  F  (  x  )  =      ∫                  x        3                            x        6              (  2  t  −  1      )    3    d  t

Feriheashrz · Accepted Answer

See, Using FTC  F  (  x  )  =      ∫                  x        3                            x        6              (  2  t  −  1      )    3    d  t  =      ∫          0                      x        6              (  2  t  −  1      )    3    d  t  −      ∫    0                  x        3              (  2  t  −  1      )    3    d  t                              F          ′                (        x        )                            =                  d                      d            x                                    [                      ∫                          0                                                      x                6                                              (          2          t          −          1                      )            3                    d          t          −                      ∫            0                                          x                3                                              (          2          t          −          1                      )            3                    d          t          ]                                                  =        (        2                  x          6                −        1                  )          3                ⋅        6                  x          5                −        (        2                  x          3                −        1                  )          3                ⋅        3                  x          2                                                  =        6                  x          5                (        2                  x          6                −        1                  )          3                −        3                  x          2                (        2                  x          3                −        1                  )          3                    Using Leibniz rule  F  (  x  )  =      ∫                  x        3                            x        6              (  2  t  −  1      )    3    d  t                              F          ′                (        x        )                            =                  d                      d            x                                    ∫                                    x              3                                                          x              6                                      (        2        t        −        1                  )          3                d        t                                          =        (        2                  x          6                −        1                  )          3                ⋅        6                  x          5                −        (        2                  x          3                −        1                  )          3                ⋅        3                  x          2

Montenovofe · Accepted Answer

By fundamental theorem of calculus and chain rule  F  (  x  )  =      ∫                  x        3                            x        6              (  2  t  −  1      )    3    d  t        F    ′    (  x  )  =            [              ∫                              x            3                                                x            6                              g      (      t      )      d      t      ]        ′    ,  g  (  t  )  =  (  2  t  −  1      )    3      =            [      G      (              x        6            )      −      G      (              x        3            )      ]        ′    =  (      x    6        )    ′    g  (      x    6    )  −  (      x    3        )    ′    g  (      x    3    )    =  6      x    5    (  2      x    6    −  1      )    3    −  3      x    2    (  2      x    3    −  1      )    3

Find the derivative of the following function using the Fundamental Theorem of Calculus: F (

Answered question

Answer & Explanation

New Questions in Integral Calculus