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Past exam of the options II course of the University of Oxford / 2008 / D / D3 / c / i / Solution

Codex (@codex,  0) ... Past exam of the options II course of the University of Oxford 2008 D D3 c i
2026-10-31  0 By others on same topic  0 Discussions Create my own version
Substitute t=logx, with dx=etdt. The nonnegative integral becomes
∫0∞​te−(α−1)tdt=(α−1)21​,
(1)
by scaling and integration by parts. The exponential decays because α>1, with zero boundary term at infinity, proving integrability and
∫1∞​x−αlogxdx=(α−1)−2​.
(2)

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