 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2014/paper-1/15/b/solution

Translate the variables by

u = x - x_{0}

v = y - y_{0}

w = z - z_{0}

. The numerator becomes

u + x_{0}

. The part odd in

u

integrates to zero, while the constant part factors into three Gaussian integrals. Thus

I = x_{0} (\int_{- \infty}^{\infty} e^{- u^{2} / a^{2}} d u)^{3} = x_{0} (a π)^{3} = π^{3/2} a^{3} x_{0} .

The absolute integrability of a polynomial times a Gaussian justifies this separation by Fubini's theorem. Neither

y_{0}

nor

z_{0}

affects the result; when

x_{0} = 0

, odd symmetry makes the integral zero.

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