Spot The Flaw

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Many of you must have seen numerous examples of such situations where we get an absurd result and have to spot flaw in proof.

A famous one is the “algebraic proof of 1=0

Here I came across a “Calculus based proof of 1=0” in A Mathematical Mosaic which I wanted to share with you.

It is based on “Integration by Parts” formula, which is:

\[\int f \ dg = fg - \int g \ df \nonumber\]

So, the proof is of this is as follows:

\[\int \frac{1}{x} dx = \left(\frac{1}{x}\right) x - \int x \ d \left(\frac{1}{x}\right) = 1 - \int x \left(-\frac{1}{x^2}\right) dx = 1 + \int \frac{1}{x}dx \nonumber\] \[\Rightarrow 0 = 1 \nonumber\]