Expert: Scotto - 5/4/2009

Question
If y = exp(-0.5x^2), what's the integral of y with respect to x? In general, what's the integral of y = exp[f(x)]?

Answer
There is no way to do this integral with finite limits.
However, if the integral is from -∞ to ∞, put it in x.
Multiply by another integral in y from -∞ to ∞.
Using the x-y corrdinate system, conver this to polar coordinates.
Note that when you convert
⌠⌠
⌡⌡ f(x,y) dx dy to polar coordinates, you get
⌠⌠
⌡⌡ r f(r,Θ) dr dΘ.

If f(x,y) = exp(-x²/2)exp(-y²/2), combine the terms by adding the exponets.  The result is f(x,y) = exp(-(x²+y²)2).
Converting this to polar coordinates gives f(r,Θ) = exp(-r²/4).

When the integral is converted to polar coordinates, the result is given by the double integral with A=2π.
⌠A⌠∞
⌡0⌡0 r f(r,Θ) dr dΘ =
⌠A⌠∞
⌡0⌡0 r exp(-r²/4). Let u = r², then du = 2r dr, or (1/2)du = r dr.
When r = 0, u = 0.  When r = ∞, u is ∞.  The integral is then
⌠A⌠∞
⌡0⌡0  exp(-u/4) du dΘ.
The integral is then -4exp(-u/4) with u at ∞ - u at 0,
which is
⌠A
⌡0 4 dΘ, which is 4A.  A was 2π, so the value is 8π
when the integral is over the whole space.

That is why the normal distribution has a 1/√(2π) in it, since the value of 2π is the result of this interal².


In general, there is no formula for the integral of f(x).
However, there are books published on the integral given the form of f(x).  For example, the integral of a constant K with respect to x is Kx.  The integral of Kx^n with respect to x is [Kx^(n+1])/n+1).
The integral of Asin(Bx) with respect to x is -Acos(Bx)/B.  There are hundreds of such formula, but no specifc way to do the problems in general.