Is there any elementary function whose antiderivative contains an exact constant?Let's say we have F ( x ) = ∫ f ( t ) d t. Now it's obvious to me why F is the class of functions whose derivative yields f(x). However, I was curious if it is possible for the antiderivative to be something such as F ( x ) = tan ⁡ ( x ) + 5 + C as an example. I can see how we would merge this to become just F ( x ) = tan ⁡ ( x ) + C but I'm wondering if it is possible for an elementary antiderivative to contain a constant that the entire class of antiderivatives share. My feeling is that this would never happen but I can't seem to figure out exactly why.

Question

Is there any elementary function whose antiderivative contains an exact constant?Let&#039;s say we have   F  (  x  )  =  ∫      f    (    t    )       d  t. Now it&#039;s obvious to me why F is the class of functions whose derivative yields f(x). However, I was curious if it is possible for the antiderivative to be something such as   F  (  x  )  =  tan  ⁡  (  x  )  +  5  +  C as an example. I can see how we would merge this to become just   F  (  x  )  =  tan  ⁡  (  x  )  +  C but I&#039;m wondering if it is possible for an elementary antiderivative to contain a constant that the entire class of antiderivatives share. My feeling is that this would never happen but I can&#039;t seem to figure out exactly why.

billyfcash5n · Accepted Answer

Explanation:No function at all has an antiderivative that contains an exact constant. This is because if you add a constant to a function, no matter what it is, the function&#039;s derivative does not change, since   (  f  (  x  )  +  g  (  x  )      )    ′    =      f    ′    (  x  )  +      g    ′    (  x and the derivative of any constant is identically zero.

Is there any elementary function whose antiderivative contains an exact constant? Let's say we have

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