Apply the first fundamental theorem of calculus to each function over the interval indicated. f ( x ) = x e x over [0,2]

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Apply the first fundamental theorem of calculus to each function over the interval indicated.  f  (  x  )  =  x      e          x       over [0,2]

kartonaun · Accepted Answer

Let f be a continuous real-valued function defined on aclosed interval[a, b]. Let F be a function such that,for all x in [a, b],f(x)=F&#039;(x)Then, for all x in [a, b],  F  (  x  )  =      ∫          a              x        f  (  t  )  d  t  +  F  (  a  )and  f  (  x  )  =      d          d      x            ∫          a              x        f  (  t  )  d  tover   [  0  ,  2  ]  ⇒  f  (  x  )  =      F    ′    (  x  )  ∗  ⇒  F  (  x  )  =      ∫          a              x        f  (  t  )  d  t  +  F  (  a  )  F  (  x  )  =      ∫          0              x        x      e          x        d  x  +  F  (  0  )      ∫          0              2        x      e          x        d  x  =  x      e          x        −      e          x            )          0              2        =      e          x        (  x  −  1      )          0              2        =      e          2        (  2  −  1  )  −      e          0        (  0  −  1  )  =      e          2        +  1

Apply the first fundamental theorem of calculus to each function over the interval indicated. f(x)=xe^x over [0,2]

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