How do you use the Fundamental Theorem of Calculus to find the derivative of ∫ 1 1 + t 2 d t from x to 5?

Question

How do you use the Fundamental Theorem of Calculus to find the derivative of       ∫                  1                    1        +                  t                      2                                      d      t       from   x to   5?

Zackery Harvey · Accepted Answer

d              d        x                                 ∫        x        5                    1              1        +                  t          2                                 d      t        =    -          1              1        +                  x          2                    The FTOC tells us that:            d              d        x                                 ∫        a        x                         f              (        t        )                         d      t        =          f              (        x        )             for any constant       a  (ie the derivative of an integral gives us the original function back).We are asked to find:      E    =          d              d        x                                 ∫        x        5                    1              1        +                  t          2                                 d      t      (notice the upper bounds of the first integral are not in the correct format for the FTOC to be applied, directly). We can manipulate the definite integral using integral properties:      E    =          d              d        x                   -                  ∫        5        x                    1              1        +                  t          2                                 d      t                      =    -          d              d        x                                 ∫        5        x                    1              1        +                  t          2                                 d      t      We can now apply the FTOC to get;      E    =    -          1              1        +                  x          2

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