Explanation how it can be that f ′ ( x ) = g ( x ) but wolfram alpha says ∫ g ( x ) ≠ f ( x )?I've just started learning about antiderivatives/primitive functions/indefinite integrals, and I have the functions f ( x ) = 3 ln ⁡ ( ( x + 2 3 ) 2 + 1 ) g ( x ) = 2 x + 2 3 ( x + 2 3 ) 2 + 1 I came to the conclusion that f ′ ( x ) = g ( x ) so that ∫ g ( x ) = f ( x ) and I wanted to check with wolfram alpha, but wolfram says that ∫ g ( x ) ≠ f ( x ) even though it says f ′ ( x ) = g ( x ).It seems to me that this violates the definition of antiderivative?

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Explanation how it can be that       f    ′    (  x  )  =  g  (  x  ) but wolfram alpha says   ∫  g  (  x  )  ≠  f  (  x  )?I&#039;ve just started learning about antiderivatives/primitive functions/indefinite integrals, and I have the functions  f  (  x  )  =  3  ln  ⁡  (  (            x      +      2        3        )    2    +  1  )  g  (  x  )  =                    2                              x            +            2                    3                            (                              x            +            2                    3                          )          2                +        1            I came to the conclusion that       f    ′    (  x  )  =  g  (  x  ) so that   ∫  g  (  x  )  =  f  (  x  )  and I wanted to check with wolfram alpha, but wolfram says that   ∫  g  (  x  )  ≠  f  (  x  ) even though it says       f    ′    (  x  )  =  g  (  x  ).It seems to me that this violates the definition of antiderivative?

scoseBexgofvc · Accepted Answer

Step 1The function f(x) is one of the infinite number of antiderivatives of the function g(x) which differ only by a constant. In the case of f(x), that constant, typically denoted as C, is 0. When you integrate g(x), you get a whole set of functions. Not just one function. That set is usually denoted   f  (  x  )  +  C where   C  ∈      R  . Therefore, this means that there are going to be as many functions in that set as there as real numbers - that is, an infinite number. That&#039;s why, technically speaking,   ∫  g  (  x  )    d  x  ≠  f  (  x  ). The result of the integration process is not a function, but a set of functions. Those are slightly different concepts.

Fletcher Hays · Accepted Answer

Explanation:The symbol   ∫  g  (  x  )  d  x means the set of all antiderivatives of g. Since f is an antiderivative of g , we have   ∫  g  (  x  )  d  x  =  f  (  x  )  +  C  .

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