Another Proof Of Zero Equals One
The first experience with calculus is usually quite memorable for most of us. For some it introduces a tool of immense beauty and seemingly infinite potential to solve problems — like a super power. For others, it seems like a lifetime of suffering and madness. The reason is simple — there just too many traps and pitfalls in calculus to worry about, and even seasoned mathematicians are sometimes found scratching their heads.
In this article, we are going to see a demonstration of how things can go horribly wrong if we are not careful, while dealing with even the simplest of integrals.
Consider the following simple integral -
To solve this, we are going to use a well known formula of integration by parts, which is given by
For our case, we choose u=1/x and v=1, so that
If we look at this solution casually, there seems to be nothing wrong, and yet we know it is impossible!
There is a little trap here, that we overlooked.
The problem with our solution is that we ignored one key requirement of integral calculus — We should not solve integrals without its proper limits, otherwise we might not get a sensible result.
Let us now repeat the calculations, taking some arbitrarily chosen proper limits. So, this time we have
so that we get,
and all is well! So, the moral of the story is — Always choose some limits while doing integrals.
Originally published at https://physicsgarage.com on April 13, 2020.
