 Solution

ID: past-exam-of-the-mathematical-methods-ii-course-of-the-university-of-oxford/2014-resit/4/b/solution

On bounded intervals the exponential series converges uniformly, allowing termwise integration. Since

e^{- t^{2}} = 1 - t^{2} + t^{4} /2 + O (t^{6})

I (x) = \frac{2}{π} (x - \frac{x ^{3}}{3} + \frac{x ^{5}}{10} + O (x^{7})) .

The requested first two nonzero terms are therefore

I (x) = \frac{2 x}{π} - \frac{2 x ^{3}}{3 π} + O (x^{5}) .

I (0) = 0

2 e^{- x^{2}} / π

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