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

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- Solution 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

is

(lo g R)^{2} /8

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

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