 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2007/paper-1/section-b/3/a/i/solution

Since the derivative of the arctangent is

1/ (1 + x^{2}) = 1 - x^{2} + O (x^{4})

near zero, termwise integration gives

arctan x = x - \frac{x ^{3}}{3} + O (x^{5}) .

The integration constant is zero because

arctan 0 = 0

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