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The condition for functions to be odd or even are
\[f(x)=-f(-x)\]

\[f(x)=f(-x)\]

respectively.
When you differentiate these equations you get
\[f'(x)=--f'(-x)=f'(-x)\]
 (1)
\[f'(x)=-f'(-x)\]
 (2)
Differentiating an odd function gives an even function and vice versa. The same is not true when integrating in general, because of the role of the arbitrary constant .Integrating(1)and(2)gives
\[f(x)=-f(-x)+c\]

\[f(x)=f(-x)+c\]

so the odd and even conditions fail without the further condition that  
\[c=0\]
.