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TFM
#87
Feb9-09, 03:24 PM
P: 1,031
Okay so:

[tex] <x^2> = \int^{\infty}_0 x^2 P(x) [/tex]

P(x) = B^2 x e^{-2\beta x}

[tex] <x^2> = \int^{\infty}_0 x^2B^2 x e^{-2\beta x} [/tex]

[tex] <x^2> = \int^{\infty}_0 x^3B^2 e^{-2\beta x} [/tex]

[tex] <x^2> = B^2 \int^{\infty}_0 x^3e^{-2\beta x} [/tex]

I have a feeling that this integral will have to be done three times...

Okay so first time:

[tex] \left[\int^{\infty}_0 x^3e^{-2\beta x}\right] [/tex]

[tex] f(x) = x^3, f'(x) = 3x^2 [/tex]

[tex] g'(x) = e^{-2\beta x}, g(x) = -\frac{1}{2\beta}e^{-2\beta x} [/tex]

Thus:

[tex] \left[ -x^3\frac{1}{2\beta}e^{-2\beta x} - \int -3x^2\frac{1}{2\beta}e^{-2\beta x} \right] [/tex]

[tex] \left[ -x^3\frac{1}{2\beta}e^{-2\beta x} + \frac{3}{2\beta}\int x^2e^{-2\beta x} \right] [/tex]

Okay so far?