Can't do this integral...

[daon]

What tools should I apply to an integral in the form:

Int(e^(x^2))dx

I have tried everything I know of, and they won't handle this..

 

[Matt]

It can't be done.

 

[daon]

Why is that? Theoretically, it should be able to be done since there is an area under the curve of e^(x^2). Having tried nearly every calc technique I could think of, I guess I know "why", but is it because someone hasn't found the proper method?

If I use mathematica to find this integral it outputs a bunch of constants and a function Erfi(). What exatly is this function? Why can't the integral be done to reflect this function?

 

[pka]

When Matt says it cannot be done, he means that there is no elementary closed form antiderivative of \(\displaystyle \L
e^{x^2 }\) .
That is, there is no simple function the derivative of which is \(\displaystyle \L
e^{x^2 }\) .

You are quite correct in saying that over any compact interval the integral of that function represents the area under the graph on the interval.

In order to find that we use the series representation: \(\displaystyle \L
e^{x^2 } = \sum\limits_{k = 0}^\infty {\frac{{x^{2k} }}{{k!}}}\) .

That series can be integrated term by term and will converge for all x.

 

[daon]

When Matt says it cannot be done, he means that there is no elementary closed form antiderivative of \(\displaystyle \L
e^{x^2 }\) .
That is, there is no simple function the derivative of which is \(\displaystyle \L
e^{x^2 }\) .

You are quite correct in saying that over any compact interval the integral of that function represents the area under the graph on the interval.

In order to find that we use the series representation: \(\displaystyle \L
e^{x^2 } = \sum\limits_{k = 0}^\infty {\frac{{x^{2k}}}{{k!}}}\) .

That series can be integrated term by term and will converge for all x.

So to compute the integral of f'(x) = e^(x^2), we can compute the integral of that summation? But how do we find a generalized solution not given any boundires for x like f(x,c)?

Can we do this:
\(\displaystyle \L
\int_{x1}^{x2} e^{x^2 }dx = \int_{x1}^{x2}\sum\limits_{k = 0}^\infty {\frac{{x^{2k} }}{{k!}}}dx\)
Is this as far as I can go without knowing specific values for x?

 

[tkhunny]

What tools should I apply to an integral in the form:

Int(e^(x^2))dx

I have tried everything I know of, and they won't handle this..

To answer your question directly and literally,

1) Series Methods, as shown by pka.
2) Numerical Methods.

In other matters:

"Theoretically, it should be able to be done since there is an area under the curve of e^(x^2)."

At some point, you will need to abandon this elementary understanding of the integral. Proof of existence is not nearly the same as demonstration of value.

"Having tried nearly every calc technique I could think of"

Can you imagine how much mathematics you've never even HEARD of? The breadth of experience of ANY living individual probably is not significant. Some historians refer to an individual at the turn of the 20th century as the LAST human to be conversant in ALL of mathematics. There is just too much now. You can't know enough. Thanks for trying, though.

"is it because someone hasn't found the proper method?"

The answer to this has disappointed students of mathematics for generations, to discover that the real world is not so neat and clean as the examples in the book or the homework problems. As the two listed above, there are whole branches of mathematics at one's disposal so that one can deal with such problems. There are remarkably simple things that have no convenient solution. Try solving this, \(\displaystyle x+2\,=\,e^{x}\) for x, for example.

Disclaimer - I'm not trying to put you down. Frankly, I'm impressed that you got to the point in your mathematics education where you would be asking such a question. Not everyone reaches that level of thought. Go right ahead and keep asking questions that hold interest for you. You will learn much more than if you just sit back and hope to absorb everything. You will learn much more than if you memorize everything you need to work all the homework problems. Try not to be offended when you don't like the answer. It is very likely that someone before you has asked the same question.

 

[pka]

Well actually, one would approximation techniques for the sums.

We get \(\displaystyle \L
\left. {\sum\limits_{k = 0}^\infty {\frac{{x^{2k + 1} }}{{\left( {2k + 1} \right)\left( {k!} \right)}}} } \right|_{x_1 }^{x_2 }\).

But of course, almost any advanced calculator or CAS will do the definite integral with a great deal of accuracy.

 

[daon]

tk, I appreciate the critisism, and although it may be impossible for me to fluently understand all mathematics, I intend to try. Math (along with computer science and physics) has always been a hobby of mine and as an undergraduate, I just decided to persue it, though I'm not sure what feild. It doesn't bother me that this particular problem may not have a definite answer, but I still will try to find one until I decide for myself it is not possible. That is, until the proof of the "answer's" non-existance makes sense to me. I guess I have always been that way.

pka, Thanks, you have been a big help. I even learned a little latex, hah. Can you tell me exactly how you approximated the integral by altering the summation? If you just used the power rule, is there any way we can get a general solution?