Must a Function be Bounded for the Antiderivative to Exist over a Given Interval?In my Calculus class, we were given the following definition of antiderivative:Let f be a function defined on an interval.An antiderivative of f is any function F such that F ′ = f.The collection of all antiderivatives of f is denoted ∫ f ( x ) d x .My question is, don't we have to say that f should be bounded in the definition?If not, then f is not integrable by definition, so we can't say anything about ∫ f ( x ) d x , right?

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Must a Function be Bounded for the Antiderivative to Exist over a Given Interval?In my Calculus class, we were given the following definition of antiderivative:Let f be a function defined on an interval.An antiderivative of f is any function F such that       F    ′    =  f.The collection of all antiderivatives of f is denoted       ∫    f    (    x    )    d    x  .My question is, don&#039;t we have to say that f should be bounded in the definition?If not, then f is not integrable by definition, so we can&#039;t say anything about       ∫    f    (    x    )    d    x  , right?

pompatzz8 · Accepted Answer

Step 1No, the function does not have to be bounded to have an integral.Consider       ∫    0    1              d      x              x      which is an improper integral because the integrand is not bounded on (0,1).Step 2However the anti derivative is   2      x   which results in a bounded value for       ∫    0    1              d      x              x        =  2.

Gauge Terrell · Accepted Answer

Step 1The existence of an antiderivative and being integrable are distinct (although related) concepts.Take                     f        :                    [        0        ,        1        ]                    ⟶                              R                                                x        ↦                              {                                                                      x                  2                                sin                ⁡                                  (                                      1                                          x                      2                                                        )                                                             if                 x                &amp;gt;                0                                                                    0                                             if                 x                =                0.                                                                  Step 2Then f is differentiable, but f′ is unbounded. But, in particular, f is an antiderivative of f′.

Must a Function be Bounded for the Antiderivative to Exist over a Given Interval? In my Calculus cl

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