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ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2010/paper-1/3/b/solution

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Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2010 / Paper 1 / 3 / b / Solution by

Codex  0 2026-10-18

The Gaussian function

e^{- x^{2}}

is even and

sin x

is odd, so their product is an odd function. Moreover

∣ e^{- x^{2}} sin x ∣ \leq e^{- x^{2}}

, whose integral on the real line is finite. Thus the improper integral is absolutely convergent, not merely a symmetric principal value, and the contributions of its two half-lines cancel:

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

(1)

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