I am a little bit confused about how the 'integration by parts' formula actually works.
My problem is as follows:
The integration by parts rule is device from the product rule -> (fg)' = f'g + fg'
Then by integrating throughout you get this (using S as integral symbol):
fg = S f'g dx + S fg' dx
My confusion relates to the fact that this is not the integration by parts formula and it appears logical to me that it should be. The integration by parts formula goes one step further:
S fg' dx = fg - S f'g dx
This does not make sense to me since it is the integral of (fg)' that you want to s

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If you were trying to find the antiderivative of ∫uv' then this would make sense to me

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