 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2016/paper-1/15/a/ii/solution

(1 + x^{2})^{- 1}

. Expand it as a geometric series and integrate term by term near zero, using

arctan 0 = 0

arctan x = \int_{0}^{x} (1 - t^{2} + t^{4} + O (t^{6})) d t = x - \frac{x ^{3}}{3} + \frac{x ^{5}}{5} + O (x^{7}) .

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