Let's Prove that 1=01=0

Calculus Level 5

Read through all the steps below carefully:

  1. We know from the product rule that ddx(uv)=dvdxu+dudxv\frac{d}{dx}(uv)=\frac{dv}{dx}u+\frac{du}{dx}v.
  2. Integrate both sides with respect to xx to get uv=u dv+ v du    u dv=uvv duuv=\int u\ dv+\int\ v\ du\implies \int u\ dv=uv-\int v\ du.
  3. Consider 12dxx\int_1^2 \frac{dx}{x}.
  4. Define u=1xu=\frac{1}{x} and dv=dxdv=dx.
  5. Then, du=dxx2du=-\frac{dx}{x^2} and v=xv=x.
  6. Substitute the variables into the equation in step 2 to get 12dxx=1+12dxx\int_1^2 \frac{dx}{x}=1+\int_1^2 \frac{dx}{x}.
  7. Subtract both sides of the equation in step 6 by 12dxx\int_1^2 \frac{dx}{x} to get 0=10=1.

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

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