Performing an integral from -infinity to infinity in Maple

Question

Hey everyone,

I'm trying to perform the following integral in Maple:

simplify(int(a*x^2*e^(-a*x^2), x = -infinity .. infinity))

But instead of returning an answer, Maple just returns back the integral statement itself:

int(a*x^2*e^(-a*x^2), x = -infinity .. infinity)

In the prettier form though (with the actual integral sign, etc). I've tried removing the "simplify()" but it doesn't make any difference.

Any idea why that is? It should return a value.

Thanks!

Amit

Answer 1

For what it's worth Wolfram Alpha seems to be able to do this: http://www.wolframalpha.com/input/?i=integrate%28a*x%5E2*e%5E%28-a*x%5E2%29%2C+x+%3D+-infinity+..+infinity%29

It gives an answer of sqrt(pi) / (2 * sqrt (a)) (assuming Re(a) > 0).

Answer 2

You have to tell maple that a has positive real part. Otherwise, the integral does not exist. Consult the manual for how to do this.

IIRC, there is a assume function, and a symbolic option you can pass to various functions (including simplify and int).

Answer 3

You may have just a syntax problem. The exponential function applied to x is exp(x) not e^x.

> int(a*x^2*exp(-a*x^2), x = -infinity .. infinity);
                /               (1/2)           \
                |             Pi                |
       piecewise|csgn(a) = 1, --------, infinity|
                |                (1/2)          |
                \             2 a               /