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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).

References

[1] E. W. Weisstein, "Gaussian Integral," Wolfram MathWorld. (Jan 25, 2016) mathworld.wolfram.com/GaussianIntegral.html.

External Links

Permanent Citation

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