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Calculus/Non Integrable functions

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Expert: - 5/5/2009


Question
I tried much to find the value of the integration of (e^X^2)

Answer
There is no way to integrate e^(x²).

However, if you're integrating e^(-x²) from -∞ to ∞, do the following:

Multiply two integrals with this value together.  The result is
⌠⌠
⌡⌡ e^(-x²-y²) dx dy.

Convert that to polar coordinates.  Note that
⌠⌠
⌡⌡ f(x,y) dx dy =
⌠⌠
⌡⌡ r f(r,Θ) dr dΘ.
The limits on the first one go from -∞ to ∞ for both x and y.
The limits on the second one go from 0 to ∞ for r and 0 to 2π for Θ.
Also, to convert to polar, note that r² = x²+y² and
Θ = (inverse tan)(x/y), being put in the quadrant where x and y are.

That is now we have
⌠⌠
⌡⌡ r e^-r² dr dΘ.

The answer is just e^r²/2 when the dr is done.
It is then 2πe^-r² when Θ is done.
The original integral was squared to do this,
so we have the value as √(2π) e^(-r²/2).
This is where the normal distribution comes from.

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