TMUA Yotta Paper 2 · Question 9

It is given that $\int u\frac{\dd v}{\dd x}\,\dd x=uv-\int v\frac{\dd u}{\dd x}\,\dd x$ . Here is an attempt to prove $0=1$ :

Let $u=\frac1x$ and $\frac{\dd v}{\dd x}=1$
$\frac{\dd u}{\dd x}=-\frac1{x^2}$ and $v=x$
$\int\frac1x\,\dd x=1+\int\frac1x\,\dd x$
$\cancel{\int\frac1x\,\dd x}=1+\cancel{\int\frac1x\,\dd x}$ . Therefore, $0=1$ . QED.

AThe proof is incorrect, and the first error is on line 1.

BThe proof is incorrect, and the first error is on line 2.

CThe proof is incorrect, and the first error is on line 3.

DThe proof is incorrect, and the first error is on line 4.

EThe proof is fully correct.

examday