 Solution

ID: past-exam-of-the-options-ii-course-of-the-university-of-oxford/2009/d/d3/i/a/solution

The function is nonnegative and measurable, so Lebesgue integrability is equivalent to finiteness of its integral. For

α \neq = - 1

\int_{ε}^{1} x^{α} d x = \frac{1 - ε ^{α + 1}}{α + 1} .

Letting

ε ↓ 0

gives a finite limit exactly when

α > - 1

; at

α = - 1

the integral is

- lo g ε

α > - 1 .

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