# Integrals that can't be evaluated

#### lvlastermind

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.

#### slider142

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?

#### lvlastermind

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...

#### slider142

Since it is Riemann integrable on every interval not containing 0, let
$$F(x) = \int_0^x (1 + \frac{1}{t})^t dt$$
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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#### lvlastermind

Since it is Riemann integrable on every interval not containing 0, let
$$F(x) = \int_0^x (1 + \frac{1}{t})^t dt$$
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?

#### HallsofIvy

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!

#### lvlastermind

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...

#### lvlastermind

This is the last time I'll bump this thread and if no one has an answer I'll let it die...

#### HallsofIvy

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".

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