Past exam of the options I course of the University of Oxford / 2010 / D / D2 / 1

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- Solution 1

Solution

Both functions are integrable on $(0, 1)$ . The oscillatory function is measurable and bounded by

1

in absolute value, on an interval of finite measure. For the other function,

\int_{0}^{1} h (x) d x = \int_{0}^{1} \frac{d x}{1 - x ^{2}} = \frac{π}{2} < \infty.

In particular its singularity at the right endpoint is of integrable inverse-square-root type.

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