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If one would want to prove that the indefined integral :

[itex]\int[f(x)+g(x)]dx[/itex] = [itex]\int f(x)dx[/itex] + [itex]\int g(x)dx[/itex].

Would this be apropriate:

A(x) = [itex]\int[f(x)+g(x)]dx[/itex];

B(x) = [itex]\int f(x)dx[/itex];

C(x) = [itex]\int g(x)dx[/itex].

And since the primitive of a fuction is another fuction whose derivative is the original fuction:

A'(x) = f(x) + g(x);

B'(x) + C'(x) = (B + C)'(x) = f(x) + g(x).

What would imply bt the Mean Value Theorem: A(x) = B(x) + C(x) + K.

Is this a approiate proof?

If so, who do we know that k = 0? Since there is no K in " [itex]\int[f(x)+g(x)]dx[/itex] = [itex]\int f(x)dx[/itex] + [itex]\int g(x)dx[/itex]."

Regards,

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# Primitives, Proof based on theorems for differentiation

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