 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2021/paper-2/4/solution

For

n = 0

s = a x

reduces the integral to the given Gaussian integral:

\int_{- \infty}^{\infty} e^{- a x^{2}} d x = \frac{1}{a} \int_{- \infty}^{\infty} e^{- s^{2}} d s = \frac{π}{a} .

For

n = 1

, the integrand is an integrable odd function, so its contributions from the two half-lines cancel:

\int_{- \infty}^{\infty} x e^{- a x^{2}} d x = 0.

The condition

a > 0

ensures convergence in both cases.

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