Some Gaussian Integrals

n

0

1

2

3

b

1

2

3

4

5

p

1

p

∫

0

n

x

-b

2

x

e

dx = 0.31606

The well-known Gaussian integral

∞

∫

0

-

2

x

e

dx=

π

2

can be evaluated in closed form, even though there is no elementary function equal to the indefinite integral

∫

-

2

x

e

dx

. More generally, integrals of the form

∞

∫

0

n

x

-b

2

x

e

dx

can be evaluated for positive integers

n

[1]. In this Demonstration, we perform numerical integrations for

p

∫

0

n

x

-b

2

x

e

dx

, with

n=0,1,2,3

and

b=1,2,…,5

. The results are then represented as areas under a curve (shown in red).