Past exam of the options II course of the University of Oxford / 2009 / D / D3 / ii / a

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- Solution a

Solution

The function is measurable and negative on this interval. Its absolute value obeys

0 \leq \frac{- x l o g x}{( 1 + x ) ^{2}} \leq - x lo g x, \int_{0}^{1} - x lo g x d x = \frac{1}{4} .

The final integral follows by integration by parts and

x^{2} lo g x \to 0

at zero. The comparison test for integrals proves integrability.

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