How can [x] be moved into the integrand in the ∫ x^2 d[x] equation?

In summary, the conversation discusses a question from a book regarding integration with respect to the greatest integer function of x. The participants suggest using the equation d[x]=Ʃ_n δ(x-n) dx or integration by parts to solve the integral ∫ x^2 d[x] from -5 to 7.
  • #1
kushan
256
0
I was going through a book
Which had this question
∫ x^2 d[x] from -5 to 7

Which means integration wrt to greatest integer function of x
Any idea on how to go about it
 
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  • #2
kushan said:
I was going through a book
Which had this question
∫ x^2 d[x] from -5 to 7

Which means integration wrt to greatest integer function of x
Any idea on how to go about it

d[x]=Ʃ_n δ(x-n) dx as far as I can see it ... or maybe integration by parts is less ambiguous
 
Last edited:
  • #3
Can you please explain more
 
  • #4
∫xdy=xy-∫ydx, you can move [x] into the integrand
 

1. What is the Integration Floor function?

The Integration Floor function, denoted as ⌊ x ⌋, is a mathematical function that rounds down a given real number to the nearest integer. This function is commonly used in calculus and discrete mathematics.

2. How is the Integration Floor function different from the regular Floor function?

While both functions round down to the nearest integer, the regular Floor function rounds down towards negative infinity, while the Integration Floor function rounds down towards zero. For example, ⌊ 3.5 ⌋ = 3, while ⌊ -3.5 ⌋ = -4.

3. What is the domain and range of the Integration Floor function?

The domain of the Integration Floor function is all real numbers, while the range is the set of all integers. This means that any real number can be input into the function, but the output will always be an integer.

4. How is the Integration Floor function used in calculus?

In calculus, the Integration Floor function is often used to find the greatest integer function, which is the largest integer that is less than or equal to a given real number. This function is used to define discontinuities and to find the values of certain integrals.

5. Can the Integration Floor function be used to solve real-world problems?

Yes, the Integration Floor function can be used to solve real-world problems. For example, it can be used in computer programming to round down a given value to the nearest integer, which can be useful in situations such as calculating the number of items to be purchased or the number of people attending an event.

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