 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2024/paper-1/17/b/solution

t = tan x

, with

d t = (1 + tan^{2} x) d x

. The endpoints become

1

and

3

, so

\int_{1}^{3} \frac{d t}{( 1 + t ) ^{2}} = [- \frac{1}{1 + t}]_{1}^{3} = \frac{1}{2} - \frac{1}{1 + 3} = 1 - \frac{3}{2} .

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