Conceptual integration question

[Krushnaraj Pandya]

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?

 

[Ray Vickson]

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 .

 

[Krushnaraj Pandya]

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 .

ah yes, I forgot all about that. thanks