 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2004/paper-1/4/a/solution

(x - 1) / [x (x + 1)] = - 1/ x + 2/ (x + 1)

. Since both endpoints are positive, integration gives

\int_{a}^{b} \frac{x - 1}{x ( x + 1 )} d x = [- ln x + 2 ln (x + 1)]_{a}^{b} = ln \frac{a ( b + 1 ) ^{2}}{b ( a + 1 ) ^{2}} .

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