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let f be a function defined on [a,b]. now, under what condition does primitive of f exists and if it exists how do we find it and its domain ?

I think, if f is continuous, then its primitive exists. But if f is not continuous on its domain, how do we know about primitive ?

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    How do you define primitive?2012-10-08

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I assume that by the primitive of $f$, you mean the antiderivative of $f$: http://en.wikipedia.org/wiki/Primitive_function.

To answer your question:

Yes, if $f$ is continuous on $[a, b]$, then the indefinite integral $F$ defined by $F(z) = \int_a^z f(x)dx$ is a primitive (or antiderivative) of $f$.

If $f$ is not continuous, $f$ may still have a primitive, but there's no general rule as to how to find it. See this page on wikipedia for more info on primitives of non-continuous functions.

Hope this helped!