Difference between integral function and antiderivative.The definition of antiderivative given in my book is:Definition:A differentiable function F(if it exists) such that F ′ = f, then F is called antiderivative of fFirst part of fundamental theorem precisely says that if a function is continuous and defined on [a,b] then integral function ∫ 0 x f ( x ) d x is differentiable and is antiderivative of f.The second part of fundamental theorem of calculus says that:Theorem: If f : [ a , b ] → R be bounded and Riemann integrable function and F be its antiderivative then ∫ a b f ( x ) d x = F ( b ) − F ( a ).My question is whether the antiderivative F given in above theorem is necessarily integral function(+some constant) i.e F ( x ) = ∫ a x f ( x ) d x + C? where C is some constant.If it is not so then give a example of bounded and Riemann integrable function whose antiderivative exists but is not equal to integral function ∫ a x f ( x ) d x

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Difference between integral function and antiderivative.The definition of antiderivative given in my book is:Definition:A differentiable function F(if it exists) such that       F    ′    =  f, then F is called antiderivative of fFirst part of fundamental theorem precisely says that if a function is continuous and defined on [a,b] then integral function       ∫    0          x        f  (  x  )  d  x is differentiable and is antiderivative of f.The second part of fundamental theorem of calculus says that:Theorem: If   f  :  [  a  ,  b  ]  →      R   be bounded and Riemann integrable function and F be its antiderivative then       ∫    a          b        f  (  x  )  d  x  =  F  (  b  )  −  F  (  a  ).My question is whether the antiderivative F given in above theorem is necessarily integral function(+some constant) i.e   F  (  x  )  =      ∫    a          x        f  (  x  )  d  x  +  C? where C is some constant.If it is not so then give a example of bounded and Riemann integrable function whose antiderivative exists but is not equal to integral function       ∫    a          x        f  (  x  )  d  x

Alisa Gilmore · Accepted Answer

Step 1A function F(x) is called an antiderivative of a function of f(x) if   F      ′    (  x  )  =  f  (  x  ) for all x in the domain of f. Note that the function F is not unique and that an infinite number of antiderivatives could exist for a given function. For example,   F  (  x  )  =      x    3  ,   G  (  x  )  =      x    3    +  5, and   H  (  x  )  =      x    3    −  2 are all antiderivatives of   f  (  x  )  =  3      x    2   because   F      ′    (  x  )  =  G      ′    (  x  )  =  H      ′    (  x  )  =  f  (  x  ) for all x in the domain of f. It is clear that these functions F, G, and H differ only by some constant value and that the derivative of that constant value is always zero. In other words, if F(x) and G(x) are antiderivatives of f(x) on some interval, then   F      ′    (  x  )  =  G      ′    (  x  ) and   F  (  x  )  =  G  (  x  )  +  C for some constant C in the interval. Geometrically, this means that the graphs of F(x) and G(x) are identical except for their vertical position.Step 2The notation used to represent all antiderivatives of a function f(x) is the indefinite integral symbol written   ∫,   ∫  f  (  x  )  d  x  =  F  (  x  )  +  C, where .The function of f(x) is called the integrand, and C is refered to as the constant of integration. The expression   F  (  x  )  +  C is called the indefinite integral of F with respect to the independent variable x. Using the previous example of   F  (  x  )  =      x    3   and   f  (  x  )  =  3      x    2  , you find that when we take an indefinite integral, we are in reality finding “all” the possible antiderivatives at once (as different values of C gives different antiderivatives)The indefinite integral of a function is sometimes called the general antiderivative of the function as well. In additionally, we would say that a definite integral is a number which we could apply the second part of the Fundamental Theorem of Calculus; but an antiderivative is a function which we could apply the first part of the Fundamental Theorem of Calculus.

Erin Lozano · Accepted Answer

Step 1Denote   G  (  x  )  =      ∫    a          x        f  (  x  )  d  xObviously   G  (  x  )  =  F  (  x  )  −  F  (  a  )(this follows from the theorem)Step 2If you now take derivatives of F and G you will see they are the same.So F and G differ by a constant.So the answer to your question is positive.

Difference between integral function and antiderivative. The definition of antiderivative given in

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