A possible way to integrate dx^2?

[depther]

While reading a proof for an integral equation I came across this formula:
http://imgur.com/K8RQlOc

It was claimed that it can be proven..

 

[andrewkirk]

If my calculations are correct, it's either meaningless or false.
It's meaningless if $dx^n$ means $(dx)^n$
It's false if $dx^n$ means $d(x^n)$. Counterexample is $n=2,\ f(x)\equiv 1$. This gives LHS=$t^2-a^2$ and RHS=$\frac{1}{2}(t-a)^2$

 

[HallsofIvy]

I wonder what "x" is supposed to represent. On the left x appears to be an n dimensional vector but on the right it is a number.

 

[depther]

Example problem provided in paper,
http://imgur.com/Cou7QoJ

 

[rs1n]

The source (of the original problem) probably uses a different convention for mutiple integration. It looks like Cauchy's formula. From wikipedia: https://upload.wikimedia.org/math/4/9/9/4996ac5454f0b1e6a964f1ec572ba146.png