 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/b/solution

t = lo g (1/ x)

, so

x = e^{- t}

and

d x = - e^{- t} d t

\int_{0}^{1} [lo g (1/ x)]^{4} d x = \int_{0}^{\infty} t^{4} e^{- t} d t = Γ (5) .

\int_{0}^{1} [lo g (1/ x)]^{4} d x = 24.

The exponential-transformed integrand also proves convergence at the logarithmically singular endpoint.

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