How to prove the function f has an antiderivative?There is a question in our analysis book and I have lots of problem with it. It says that:set f ( x ) = { 0 x ≤ 0 sin ⁡ ( π x ) x &gt; 0 g ( x ) = { 0 x ≤ 0 1 x &gt; 0. prove that f has an antiderivative but g does not.My first problem is about antiderivative of g. I think it has an antiderivative and it is G ( x ) = { 0 x ≤ 0 x x &gt; 0. Why we can't say that G is antiderivative of g? My second problem is about finding the antiderivative of f. As you may know antiderivative of sin ⁡ ( π x ) can not be shown by the elementary functions. So, to prove that f has antiderivative I can't find a function like F which its derivative is f and I need to use another approach to prove it, but I don't have any idea about what should I do?

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How to prove the function f has an antiderivative?There is a question in our analysis book and I have lots of problem with it. It says that:set   f  (  x  )  =      {                            0                                    x          ≤          0                                      sin          ⁡          (                      π            x                    )                                    x          &amp;gt;          0                          g  (  x  )  =      {                            0                                    x          ≤          0                                      1                                    x          &amp;gt;          0.                        prove that f has an antiderivative but g does not.My first problem is about antiderivative of g. I think it has an antiderivative and it is                    G        (        x        )        =                              {                                                    0                                                            x                ≤                0                                                                    x                                                            x                &amp;gt;                0.                                                                  Why we can&#039;t say that G is antiderivative of g? My second problem is about finding the antiderivative of f. As you may know antiderivative of   sin  ⁡  (      π    x    ) can not be shown by the elementary functions. So, to prove that f has antiderivative I can&#039;t find a function like F which its derivative is f and I need to use another approach to prove it, but I don&#039;t have any idea about what should I do?

sviudes7w · Accepted Answer

Step 1For f(x), use the Taylor expansion of sine about zero to express   sin  ⁡      (          π      x        )   as an infinite sum. From there, integrate across the sum to give a function for the antiderivative.Step 2As for your expression for G, the problem is that g is not defined at zero. Given that an antiderivative, by definition, has a derivative, we can say that g has no antiderivative.

Livia Cardenas · Accepted Answer

Explanation:Consider   G  (  x  )  =      ∫          −      ∞        x    g  (  t  )    d  t  .For   x  ≠  0,       G    ′    (  x  )  =  g  (  x  ), but G′(0) is not defined.

How to prove the function f has an antiderivative?

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