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 \(0=1\).

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