Integrals that can't be evaluated

  • Thread starter lvlastermind
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In summary: Almost all" integrable functions cannot be integrated in terms of functions we know simply because the set of functions we know is so small. It's not a matter of "why can we not integrate this function in terms of functions we know" but of "why can we integrate certain functions".
  • #1
lvlastermind
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This is an inquiry that has been an itch I can't reach.


Why can't the integral (1+(1/x))^x be evaluated?

I know the limit {n->inf} (1+(1/x))^x = e but I don't understand why the function doesn't have an integral.

Thanks in advance
 
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  • #2
The function is continuous and bounded everywhere but on intervals including x=0, so on any domain that does not include 0, the function is Riemann-integrable. Did you mean indefinitely integrable as a closed form expression involving a finite combination of elementary functions?
 
  • #3
slider142 said:
The function is continuous and bounded everywhere but on intervals including x=0, so on any domain that does not include 0, the function is Riemann-integrable. Did you mean indefinitely integrable as a closed form expression involving a finite combination of elementary functions?



Yes, I meant indefinitely...
 
  • #4
Since it is Riemann integrable on every interval not containing 0, let
[tex]F(x) = \int_0^x (1 + \frac{1}{t})^t dt[/tex]
Then F is an indefinite integral of your expression. Unfortunately, I do not know of any combination of elementary functions that is equivalent to F. Proving that there is no such finite combination is not a trivial matter. Have a look at http://www.soton.ac.uk/~adf/pubstore/pub20.pdf [Broken].
 
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  • #5
slider142 said:
Since it is Riemann integrable on every interval not containing 0, let
[tex]F(x) = \int_0^x (1 + \frac{1}{t})^t dt[/tex]
Then F is an indefinite integral of your expression. Unfortunately, I do not know of any combination of elementary functions that is equivalent to F.


Agreed... Hence I'm back to square one...

Anyone know?
 
  • #6
Sorry to bump this thread but I'm really interested about the answer...
 
  • #7
  • #8
The problem is "what do you mean by "can't be evaluated"? The function "1+ (1/x)x" is continuous and so certainly can be evaluated. Do you mean "evaluated in terms of elementary functions"? That's a completely different question!
 
  • #9
HallsofIvy said:
The problem is "what do you mean by "can't be evaluated"? The function "1+ (1/x)x" is continuous and so certainly can be evaluated. Do you mean "evaluated in terms of elementary functions"? That's a completely different question!

Yes, indefinite integral is what I'm looking for...
 
  • #10
This is the last time I'll bump this thread and if no one has an answer I'll let it die...
 
  • #11
Good. Your question really doesn't make much sense to begin with. "Almost all" integrable functions cannot be integrated in terms of functions we know simply because the set of functions we know is so small. It's not a matter of "why can we not integrate this function in terms of functions we know" but of "why can we integrate certain functions".
 

1. What is an integral that can't be evaluated?

An integral that can't be evaluated is a mathematical expression that cannot be calculated using standard integration techniques. These types of integrals often involve complex functions or do not have a closed-form solution.

2. How do I know if an integral can't be evaluated?

There are a few signs that may indicate an integral cannot be evaluated. One common sign is the presence of a function that does not have a known antiderivative, such as the exponential function. Additionally, if an integral has a variable limit of integration or an integrand with multiple variables, it may not be possible to evaluate it.

3. Can I still find an approximate value for an integral that can't be evaluated?

Yes, it is possible to approximate the value of an integral that cannot be evaluated using numerical integration methods such as the trapezoidal rule or Simpson's rule. These methods divide the integral into smaller, easier-to-calculate parts and sum them to approximate the overall value.

4. Are there any techniques for solving integrals that can't be evaluated?

Yes, there are some advanced techniques that can be used to solve certain types of integrals that cannot be evaluated using standard methods. These include techniques like integration by parts, substitution, and partial fractions. However, these methods may not always be successful in finding a solution.

5. How can I handle an integral that can't be evaluated in a real-world problem?

In real-world problems, it is often sufficient to find an approximate value for an integral that cannot be evaluated. Additionally, it may be possible to simplify the problem or use numerical techniques to find a solution. In some cases, it may also be necessary to seek assistance from a mathematician or computer program to find a solution.

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