 Solution

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

For

x \geq 1

(1 + x)^{2} \leq 4 x^{2}

, so

\frac{x l o g x}{( 1 + x ) ^{2}} \geq \frac{l o g x}{4 x} \geq 0.

The integral of the lower bound from one to

R

(lo g R)^{2} /8

, which diverges. Thus the nonnegative function has infinite integral and is not integrable.

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