# Integration by parts notation problem

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?

• $du=f''(t)\,dt$. – Andreas Blass Oct 20 '16 at 0:14

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 $du=f''(v)\,dt$