integration of e pow(-x2)?

Answer 1

It's a tough question, and the answer leads into what is called differential Galois theory (see my source). One says that an antiderivative F of your function is not "elementary". Elementary is a bit of a nebulous word, but one version is as follows.
F is "elementary" if it can be expressed in terms of trig functions, e^x, or polynomials by the use of addition, composition, subtraction, multiplication, division, exponentiation, or function inversion. [This may be somewhat redundant, or perhaps I could include more things. It's still true.] But you're not allowed to take an antiderivative.

Just as there is a sense in which most real numbers are irrational and transcendental, most antiderivatives are not elementary. And most power series. It's very easy to come up with elementary functions with nonelementary antiderivatives, as long as you avoid rational functions and trig polynomials.
For instance sin(x^2), 1/(log(x)), sin(x)/x, your example lack elementary antiderivatives. But it's incredibly hard to prove they are not elementary (see the reference).

It's not a big deal if a function isn't elementary. In your case you have what's called the "error function", which is a very well understood function. If you know a function's derivative and a particular value, you understand the function. See Mathworld, PlanetMath, wikipedia to see how people work with it.

I'd like to point out that before Euler, e^x would not have been counted as an elementary function. And 1/x did not have an elementary antiderivative. But then Euler *defined* log x to be the antiderivative of 1/x, inverted it, and that's where we get e^x (the "e" is for "Euler"). In some texts, sin(x) is defined as the inverse of the antiderivative of 1/sqrt(1-x^2), and one can develop trigonometry based on this.

Answer 2

This integration must be done numerically. There is no closed form to the integral.

Answer 3

integral(e^(-x^2))=?
There must be a factor in terms of x

example:
x e^(-x^2)

Answer 4

signify the integrand as e^x.dsinx: ?e^x cosx dx=e^x sinx -?sinx e^x dx +c1 we now have: -?sinx e^x dx=?e^x dcosx=e^x cosx - ?cosx e^x dx + c2 Substituting the above with later: ?e^x cosx dx = e^x sinx+e^x cosx - ?e^x cosx dx +c1+c2 for this reason: ?e^x cosx dx = a million/2 e^x (sinx+cosx) + c

Answer 5

-2x e pow(-x2)
(if it is x^2)

or

-2 e pow(-2x) (if it is -2x)