I've encountered a problem by trying to use integration by parts to prove a certain theorem. I'm not sure if this is just confusion caused by notation, or I'm missing something important. The problem is proving the next line:
$ \int_a^x f'(t)dt = tf'(t)|_a^x - \int_a^xtf''(t)dt $
Let: $u=f'(t)$, and $dv=dt$. Therefore: $du=f''(t)$, and $v=t$.
Using integration by parts:
$\int_a^x f'(t)dt =\int_a^x udv= uv|_a^x -\int_a^xvdu=tf'(t)|_a^x-\int_a^xtf''(t)$
Notice that the last integral is missing the $dt$ symbol.
Where is the mistake in my logic? Is my understanding of notation wrong? Am I missing something obvious?
See you have written $u=f'(t)$ and $t=v$. This means after your substitution $u=f'(v)$ so $du= d(f'(v))$. By chain rule you get $f''(v)\,dv$ . But $dv=dt$. Hence $du=f''(v)\,dt$