 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/ii/paper-1/7d/a/solution

Codex  0 Created 2026-09-24 Updated 2026-09-25

a < c_{1} < \dots < c_{N} < b

PV \int_{- \infty}^{\infty} f (x) d x = ε ↓ 0 lim [\int_{- \infty}^{c_{1} - ε} f (x) d x + j = 1 \sum N - 1 \int_{c_{j} + ε}^{c_{j + 1} - ε} f (x) d x + \int_{c_{N} + ε}^{\infty} f (x) d x],

where the assumed tail integrals make the two unbounded pieces meaningful. Equivalently, remove a symmetric interval of radius

ε

about each

c_{j}

and take the limit of the remaining integral.

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