A possible way to integrate dx^2?

  • Thread starter depther
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In summary, The conversation is discussing a formula from a paper for an integral equation, which the speaker claims can be proven. However, the other person points out that depending on the interpretation of ##dx^n##, the formula could be either meaningless or false. They provide a counterexample for the false case and question the meaning of "x" in the equation. It is suggested that the source may use a different convention for multiple integration.
  • #1
depther
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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..
 
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  • #2
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##
 
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  • #3
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.
 

Related to A possible way to integrate dx^2?

1. What is the purpose of integrating dx^2?

The purpose of integrating dx^2 is to find the area under a curve represented by the function f(x) = x^2. This is useful in many scientific and mathematical applications, such as calculating volumes and areas in physics and engineering.

2. How is dx^2 integrated?

To integrate dx^2, we use the power rule of integration, which states that the integral of x^n is equal to (x^(n+1)) / (n+1). In the case of dx^2, n = 2, so the integral becomes (x^3) / 3 + C, where C is the constant of integration.

3. What is the difference between integrating dx^2 and dx?

Integrating dx^2 is different from integrating dx because dx^2 represents a function with a higher degree than dx. In the case of dx, the integral is simply x + C, while dx^2 requires the use of the power rule as mentioned in the previous question.

4. Can dx^2 be integrated using other methods?

Yes, dx^2 can also be integrated using substitution, where we substitute u = x^2 and rewrite the integral in terms of u. It can also be integrated using integration by parts, although this method may require more steps and can be more complex.

5. What are the applications of integrating dx^2?

Integrating dx^2 has many applications in physics and engineering, such as calculating the work done by a force or the displacement of a moving object. It is also used in finding the area under a curve in mathematics and can be used to solve differential equations in science and engineering.

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