 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2009/paper-1/16/c/solution

p = 1/2

, use

x = t^{2}

d x = 2 t d t

, with

t > 0

. Then

x^{- 1/2} d x = 2 d t

, so

Γ (1/2) = \int_{0}^{\infty} x^{- 1/2} e^{- x} d x = 2 \int_{0}^{\infty} e^{- t^{2}} d t .

The substitution can first be made on finite positive intervals and then extended by monotone limits, justifying the endpoint passage.

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