Conceptual problem with integration by parts.

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Why is it that whenever we encounter a question which can be solved by integration by parts, we get half the function?

I mean, suppose a differentiated f(x)g(x) yielded {f'(x) g(x)dx + f(x)g'(x)dx}, then why do we get only {f'(x) g(x)dx} to extract the original function (f(x)g(x)) from?
 
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Ok...let's forget about this question for now...dont answer it.
 
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