Let's Prove that $1=0$

Read through all the steps below carefully:

1. We know from the product rule that $\frac{d}{dx}(uv)=\frac{dv}{dx}u+\frac{du}{dx}v$.
2. Integrate both sides with respect to $x$ to get $uv=\int u\ dv+\int\ v\ du\implies \int u\ dv=uv-\int v\ du$.
3. Consider $\int_1^2 \frac{dx}{x}$.
4. Define $u=\frac{1}{x}$ and $dv=dx$.
5. Then, $du=-\frac{dx}{x^2}$ and $v=x$.
6. Substitute the variables into the equation in step 2 to get $\int_1^2 \frac{dx}{x}=1+\int_1^2 \frac{dx}{x}$.
7. Subtract both sides of the equation in step 6 by $\int_1^2 \frac{dx}{x}$ to get $0=1$.

In which of these steps did I first make a mistake by using flawed logic?

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