Conceptual integration question

  • #1
Krushnaraj Pandya
Gold Member
697
73

Homework Statement


The value of ∫{[x]}dx (where {} and [] denotes the fractional part of x and greatest integer function)
1) 0
2) 1
3) 2
4)all are correct

2. The attempt at a solution
Since [x] is always an integer, the given function will always have a value=0. I thought that since the integral is a representation of the area of a curve, this would always have an area and therefore integral=0, but if we differentiate any constant we get 0 anyway, so all should be correct where is the flaw in my concept?
 

Answers and Replies

  • #2
Ray Vickson
Science Advisor
Homework Helper
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Homework Statement


The value of ∫{[x]}dx (where {} and [] denotes the fractional part of x and greatest integer function)
1) 0
2) 1
3) 2
4)all are correct

2. The attempt at a solution
Since [x] is always an integer, the given function will always have a value=0. I thought that since the integral is a representation of the area of a curve, this would always have an area and therefore integral=0, but if we differentiate any constant we get 0 anyway, so all should be correct where is the flaw in my concept?
Indefinite integrals should always have a "constant of integration". In other words, if ##F(x)## is an indefinite integral of ##f(x)##, then so is ##F(x) + C## for any constant ##C##. See, eg., https://en.wikipedia.org/wiki/Constant_of_integration .
 
  • #3
Krushnaraj Pandya
Gold Member
697
73

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