 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2013/paper-2/13/c/ii/solution

The integral converges absolutely: near zero it behaves like

lo g x

, and at infinity like

(lo g x) / x^{2}

. Let its value be

J

x = 1/ t

gives

J = \int_{0}^{\infty} \frac{- l o g t}{1 + t ^{2}} d t = - J .

Therefore

J = 0 .

Absolute convergence makes this cancellation between the two ends legitimate.

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