What is $int lfloor x^n rfloor dx$ where $n$ is any real number?
I've been trying to figure out a general rule for integrating functions of the form $lfloor x^n rfloor$. I don't have any ideas other than trying some kind of clever substitution but I have no idea what that would look like.
I know because of some theorems that $int lfloor x^n rfloor dx = x * lfloor x^n rfloor + f(lfloor x^n rfloor) + c$ but I don't know what $f$ actually looks like or whether or not $f$ has a closed form. I just know that's the form the result will look like.
integration floor-function
asked Nov 29 '18 at 3:29
The Great DuckThe Great Duck
18032047
|
show 3 more comments
I've been trying to figure out a general rule for integrating functions of the form $lfloor x^n rfloor$. I don't have any ideas other than trying some kind of clever substitution but I have no idea what that would look like.
I know because of some theorems that $int lfloor x^n rfloor dx = x * lfloor x^n rfloor + f(lfloor x^n rfloor) + c$ but I don't know what $f$ actually looks like or whether or not $f$ has a closed form. I just know that's the form the result will look like.
integration floor-function
asked Nov 29 '18 at 3:29
The Great DuckThe Great Duck
18032047
Just to clarify: $x$ is any real? or $n$ is any real? Typically, $x$ is used for reals, while $n$ is used for integers.
– JavaMan
Nov 29 '18 at 3:30
@JavaMan $x$ is a variable.
– The Great Duck
Nov 29 '18 at 3:31
That doesn't answer my question. So $n$ is a fixed real number? Or is $n$ a fixed integer? And $x$ is a variable, but presumably it is a real number?
– JavaMan
Nov 29 '18 at 3:33
@JavaMan we're integrating a function. I would believe the correct specification is that $x$ is neither? And if I said $n$ is a real number then $n$ is a real number. My choice of letter shouldn't matter to you.
– The Great Duck
Nov 29 '18 at 3:39
1
So, you want $int_{}lfloor x^nrfloor dx$ for given integer/real $n$ ? Or $int_{}lfloor x^nrfloor dn$ for given integer/real $x$?
– Jimmy R.
Nov 29 '18 at 3:43
|
show 3 more comments
I've been trying to figure out a general rule for integrating functions of the form $lfloor x^n rfloor$. I don't have any ideas other than trying some kind of clever substitution but I have no idea what that would look like.
I know because of some theorems that $int lfloor x^n rfloor dx = x * lfloor x^n rfloor + f(lfloor x^n rfloor) + c$ but I don't know what $f$ actually looks like or whether or not $f$ has a closed form. I just know that's the form the result will look like.
integration floor-function
asked Nov 29 '18 at 3:29
The Great DuckThe Great Duck
18032047
I've been trying to figure out a general rule for integrating functions of the form $lfloor x^n rfloor$. I don't have any ideas other than trying some kind of clever substitution but I have no idea what that would look like.
I know because of some theorems that $int lfloor x^n rfloor dx = x * lfloor x^n rfloor + f(lfloor x^n rfloor) + c$ but I don't know what $f$ actually looks like or whether or not $f$ has a closed form. I just know that's the form the result will look like.
integration floor-function
integration floor-function
asked Nov 29 '18 at 3:29
The Great DuckThe Great Duck
18032047
asked Nov 29 '18 at 3:29
The Great DuckThe Great Duck
18032047
edited Nov 30 '18 at 3:46
The Great Duck
asked Nov 29 '18 at 3:29
The Great DuckThe Great Duck
18032047
asked Nov 29 '18 at 3:29
The Great DuckThe Great Duck
18032047
asked Nov 29 '18 at 3:29
The Great DuckThe Great Duck
18032047
18032047
Just to clarify: $x$ is any real? or $n$ is any real? Typically, $x$ is used for reals, while $n$ is used for integers.
– JavaMan
Nov 29 '18 at 3:30
@JavaMan $x$ is a variable.
– The Great Duck
Nov 29 '18 at 3:31
That doesn't answer my question. So $n$ is a fixed real number? Or is $n$ a fixed integer? And $x$ is a variable, but presumably it is a real number?
– JavaMan
Nov 29 '18 at 3:33
@JavaMan we're integrating a function. I would believe the correct specification is that $x$ is neither? And if I said $n$ is a real number then $n$ is a real number. My choice of letter shouldn't matter to you.
– The Great Duck
Nov 29 '18 at 3:39
1
So, you want $int_{}lfloor x^nrfloor dx$ for given integer/real $n$ ? Or $int_{}lfloor x^nrfloor dn$ for given integer/real $x$?
– Jimmy R.
Nov 29 '18 at 3:43
|
show 3 more comments
Just to clarify: $x$ is any real? or $n$ is any real? Typically, $x$ is used for reals, while $n$ is used for integers.
– JavaMan
Nov 29 '18 at 3:30
@JavaMan $x$ is a variable.
– The Great Duck
Nov 29 '18 at 3:31
That doesn't answer my question. So $n$ is a fixed real number? Or is $n$ a fixed integer? And $x$ is a variable, but presumably it is a real number?
– JavaMan
Nov 29 '18 at 3:33
@JavaMan we're integrating a function. I would believe the correct specification is that $x$ is neither? And if I said $n$ is a real number then $n$ is a real number. My choice of letter shouldn't matter to you.
– The Great Duck
Nov 29 '18 at 3:39
1
So, you want $int_{}lfloor x^nrfloor dx$ for given integer/real $n$ ? Or $int_{}lfloor x^nrfloor dn$ for given integer/real $x$?
– Jimmy R.
Nov 29 '18 at 3:43
Just to clarify: $x$ is any real? or $n$ is any real? Typically, $x$ is used for reals, while $n$ is used for integers.
– JavaMan
Nov 29 '18 at 3:30
Just to clarify: $x$ is any real? or $n$ is any real? Typically, $x$ is used for reals, while $n$ is used for integers.
– JavaMan
Nov 29 '18 at 3:30
@JavaMan $x$ is a variable.
– The Great Duck
Nov 29 '18 at 3:31
@JavaMan $x$ is a variable.
– The Great Duck
Nov 29 '18 at 3:31
That doesn't answer my question. So $n$ is a fixed real number? Or is $n$ a fixed integer? And $x$ is a variable, but presumably it is a real number?
– JavaMan
Nov 29 '18 at 3:33
That doesn't answer my question. So $n$ is a fixed real number? Or is $n$ a fixed integer? And $x$ is a variable, but presumably it is a real number?
– JavaMan
Nov 29 '18 at 3:33
@JavaMan we're integrating a function. I would believe the correct specification is that $x$ is neither? And if I said $n$ is a real number then $n$ is a real number. My choice of letter shouldn't matter to you.
– The Great Duck
Nov 29 '18 at 3:39
@JavaMan we're integrating a function. I would believe the correct specification is that $x$ is neither? And if I said $n$ is a real number then $n$ is a real number. My choice of letter shouldn't matter to you.
– The Great Duck
Nov 29 '18 at 3:39
1
1
So, you want $int_{}lfloor x^nrfloor dx$ for given integer/real $n$ ? Or $int_{}lfloor x^nrfloor dn$ for given integer/real $x$?
– Jimmy R.
Nov 29 '18 at 3:43
So, you want $int_{}lfloor x^nrfloor dx$ for given integer/real $n$ ? Or $int_{}lfloor x^nrfloor dn$ for given integer/real $x$?
– Jimmy R.
Nov 29 '18 at 3:43
|
show 3 more comments
1 Answer
1
active
oldest
votes
You have to find the regions where
$k le x^n lt k+1$
for each integer $k$.
These are
$k^{1/n} le x lt (k+1)^{1/n}$.
Then replace the integral
with a sum over these regions.
In particular,
I don't see how you
can get an indefinite integral -
you have to integrate over an actual region.
(added later)
If
$I(a, b)
=int_a^b lfloor x^n rfloor dx
$,
then,
without worrying about
possible leftover intervals
at the beginning at end,
$I(a, b)
=sum_{k =lfloor a^{1/n} rfloor}^{lfloor b^{1/n} rfloor}int_{k}^{k+1} k^n dx
=sum_{k =lfloor a^{1/n} rfloor}^{lfloor b^{1/n} rfloor} k^n
$.
This is a first attempt.
There are problems
at the ends of the intervals,
but this is enough for me now.
answered Nov 29 '18 at 5:28
marty cohenmarty cohen
72.8k549128
Can you elaborate on that last sentence?
– The Great Duck
Nov 29 '18 at 5:32
So what is the final function? Does it have no closed form?
– The Great Duck
Nov 30 '18 at 3:22
add a comment |
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1 Answer
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oldest
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oldest
votes
You have to find the regions where
$k le x^n lt k+1$
for each integer $k$.
These are
$k^{1/n} le x lt (k+1)^{1/n}$.
Then replace the integral
with a sum over these regions.
In particular,
I don't see how you
can get an indefinite integral -
you have to integrate over an actual region.
(added later)
If
$I(a, b)
=int_a^b lfloor x^n rfloor dx
$,
then,
without worrying about
possible leftover intervals
at the beginning at end,
$I(a, b)
=sum_{k =lfloor a^{1/n} rfloor}^{lfloor b^{1/n} rfloor}int_{k}^{k+1} k^n dx
=sum_{k =lfloor a^{1/n} rfloor}^{lfloor b^{1/n} rfloor} k^n
$.
This is a first attempt.
There are problems
at the ends of the intervals,
but this is enough for me now.
answered Nov 29 '18 at 5:28
marty cohenmarty cohen
72.8k549128
Can you elaborate on that last sentence?
– The Great Duck
Nov 29 '18 at 5:32
So what is the final function? Does it have no closed form?
– The Great Duck
Nov 30 '18 at 3:22
add a comment |
You have to find the regions where
$k le x^n lt k+1$
for each integer $k$.
These are
$k^{1/n} le x lt (k+1)^{1/n}$.
Then replace the integral
with a sum over these regions.
In particular,
I don't see how you
can get an indefinite integral -
you have to integrate over an actual region.
(added later)
If
$I(a, b)
=int_a^b lfloor x^n rfloor dx
$,
then,
without worrying about
possible leftover intervals
at the beginning at end,
$I(a, b)
=sum_{k =lfloor a^{1/n} rfloor}^{lfloor b^{1/n} rfloor}int_{k}^{k+1} k^n dx
=sum_{k =lfloor a^{1/n} rfloor}^{lfloor b^{1/n} rfloor} k^n
$.
This is a first attempt.
There are problems
at the ends of the intervals,
but this is enough for me now.
answered Nov 29 '18 at 5:28
marty cohenmarty cohen
72.8k549128
Can you elaborate on that last sentence?
– The Great Duck
Nov 29 '18 at 5:32
So what is the final function? Does it have no closed form?
– The Great Duck
Nov 30 '18 at 3:22
add a comment |
You have to find the regions where
$k le x^n lt k+1$
for each integer $k$.
These are
$k^{1/n} le x lt (k+1)^{1/n}$.
Then replace the integral
with a sum over these regions.
In particular,
I don't see how you
can get an indefinite integral -
you have to integrate over an actual region.
(added later)
If
$I(a, b)
=int_a^b lfloor x^n rfloor dx
$,
then,
without worrying about
possible leftover intervals
at the beginning at end,
$I(a, b)
=sum_{k =lfloor a^{1/n} rfloor}^{lfloor b^{1/n} rfloor}int_{k}^{k+1} k^n dx
=sum_{k =lfloor a^{1/n} rfloor}^{lfloor b^{1/n} rfloor} k^n
$.
This is a first attempt.
There are problems
at the ends of the intervals,
but this is enough for me now.
answered Nov 29 '18 at 5:28
marty cohenmarty cohen
72.8k549128
You have to find the regions where
$k le x^n lt k+1$
for each integer $k$.
These are
$k^{1/n} le x lt (k+1)^{1/n}$.
Then replace the integral
with a sum over these regions.
In particular,
I don't see how you
can get an indefinite integral -
you have to integrate over an actual region.
(added later)
If
$I(a, b)
=int_a^b lfloor x^n rfloor dx
$,
then,
without worrying about
possible leftover intervals
at the beginning at end,
$I(a, b)
=sum_{k =lfloor a^{1/n} rfloor}^{lfloor b^{1/n} rfloor}int_{k}^{k+1} k^n dx
=sum_{k =lfloor a^{1/n} rfloor}^{lfloor b^{1/n} rfloor} k^n
$.
This is a first attempt.
There are problems
at the ends of the intervals,
but this is enough for me now.
answered Nov 29 '18 at 5:28
marty cohenmarty cohen
72.8k549128
edited Nov 29 '18 at 5:56
answered Nov 29 '18 at 5:28
marty cohenmarty cohen
72.8k549128
answered Nov 29 '18 at 5:28
marty cohenmarty cohen
72.8k549128
answered Nov 29 '18 at 5:28
marty cohenmarty cohen
72.8k549128
72.8k549128
Can you elaborate on that last sentence?
– The Great Duck
Nov 29 '18 at 5:32
So what is the final function? Does it have no closed form?
– The Great Duck
Nov 30 '18 at 3:22
add a comment |
Can you elaborate on that last sentence?
– The Great Duck
Nov 29 '18 at 5:32
So what is the final function? Does it have no closed form?
– The Great Duck
Nov 30 '18 at 3:22
Can you elaborate on that last sentence?
– The Great Duck
Nov 29 '18 at 5:32
Can you elaborate on that last sentence?
– The Great Duck
Nov 29 '18 at 5:32
So what is the final function? Does it have no closed form?
– The Great Duck
Nov 30 '18 at 3:22
So what is the final function? Does it have no closed form?
– The Great Duck
Nov 30 '18 at 3:22
add a comment |
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Just to clarify: $x$ is any real? or $n$ is any real? Typically, $x$ is used for reals, while $n$ is used for integers.
– JavaMan
Nov 29 '18 at 3:30
@JavaMan $x$ is a variable.
– The Great Duck
Nov 29 '18 at 3:31
That doesn't answer my question. So $n$ is a fixed real number? Or is $n$ a fixed integer? And $x$ is a variable, but presumably it is a real number?
– JavaMan
Nov 29 '18 at 3:33
@JavaMan we're integrating a function. I would believe the correct specification is that $x$ is neither? And if I said $n$ is a real number then $n$ is a real number. My choice of letter shouldn't matter to you.
– The Great Duck
Nov 29 '18 at 3:39
1
So, you want $int_{}lfloor x^nrfloor dx$ for given integer/real $n$ ? Or $int_{}lfloor x^nrfloor dn$ for given integer/real $x$?
– Jimmy R.
Nov 29 '18 at 3:43