Antiderivative and definite integralIf f is a continuous, real-valued function on interval [a,b], then the fundamental theorem of calculus tells us that ∫ a x f ( t ) d t = F ( x )where F(x) is antiderivative, i.e. F ( x ) ′ = f ( x ).If so, why I can't find the equality ∫ f ( x ) d x = ∫ a x f ( t ) d t = F ( x ) anywhere? It expresses the relationship between definite and indefinite integral in such a straightfoward way (assuming this equality is true). So it it true and can I use ∫ and ∫ a x interchangeably?

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Antiderivative and definite integralIf f is a continuous, real-valued function on interval [a,b], then the fundamental theorem of calculus tells us that      ∫    a    x    f  (  t  )  d  t  =  F  (  x  )where F(x) is antiderivative, i.e.   F  (  x      )    ′    =  f  (  x  ).If so, why I can&#039;t find the equality   ∫  f  (  x  )  d  x  =      ∫    a    x    f  (  t  )  d  t  =  F  (  x  ) anywhere? It expresses the relationship between definite and indefinite integral in such a straightfoward way (assuming this equality is true). So it it true and can I use   ∫ and       ∫    a    x   interchangeably?

Zemmiq34 · Accepted Answer

Step 1Actually, the Fundamental Theorem of Calculus states that if   f  :  [  a  ,  b  ]  →      R   is a continuous real-valued function, and F is its antiderivative, then      ∫    a    x    f  (  t  )  d  t  =  F  (  x  )  −  F  (  a  )Step 2This is precisely why, more accurately,      ∫    a    x    f  (  t  )  d  t  =  F  (  x  )  −  F  (  a  )  =  ∫  f  (  x  )  d  x  +  C for some   C  ∈      R

RormFrure6h1 · Accepted Answer

Step 1You are probably referring to this part of the fundemental theorem of calculus      d          d      x            ∫    a    x    f  (  t  )  d  t  =  f  (  x  )And your using the fact that   f  (  x  )  =      d          d      x        ∫  f  (  x  )  d  x which yields       d          d      x            ∫    a    x    f  (  t  )  d  t  =      d          d      x        ∫  f  (  x  )  d  x.Step 2While this is true, remember that just because two functions have the same derivative, doesn&#039;t mean that they&#039;re the same function. Here is an example                    f        (        x        )                            =                  ∫          3          x                sin        ⁡        t        d        t        =        −        cos        ⁡        x        +        cos        ⁡        3                            g        (        x        )                            =        ∫        sin        ⁡        x        d        x        =        −        cos        ⁡        x            As you can cleary see both functions have the same derivative, but the functions are not the same. Your statement will be true if a is a root of the indefinite integral. Example                               ∫          0          x                2        t        d        t        =                  x          2                −        0        =                  x          2                =        ∫        2        x        d        x            .So it really depends on the function you are using. But the statement is NOT always true.

Antiderivative and definite integral If f is a continuous, real-valued function on interval [a,b],

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