 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2013/paper-2/13/a/solution

x = 2 u

and the Gaussian being an even function:

\int_{0}^{\infty} e^{- x^{2} /4} d x = 2 \int_{0}^{\infty} e^{- u^{2}} d u = π .

The factor two from the differential exactly cancels the half-line factor in the given full-line Gaussian integral.

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