I am reaching a paradox by using integration by parts. I must be going wrong but unable to figure where I went wrong. Can you help me understand what I did wrong here?
We have two differentiable functions $ f(x) $ and $ g(x) $. Now we start with $ \int f(x) g'(x) dx $ and proceed like this:
$$ \int fg' \, dx = fg + \int f'g \, dx + C = fg + fg + \int fg'\,dx + C = 2fg + \int fg'\,dx + C. $$
What went wrong here?
Update:
As told in the first comment to this post I have made a very silly error. Here is the fix now:
$$ \int fg' \, dx = fg - \int f'g \, dx + C = fg - fg + \int fg'\,dx + C = \int fg'\,dx + C. $$
So there is no issue.
As told in the first comment to this post I have made a very silly error. Here is the fix now:
$$ \int fg' \, dx = fg - \int f'g \, dx + C = fg - fg + \int fg'\,dx + C = \int fg'\,dx + C. $$