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I'm having trouble starting with this problem. I was thinking of using the Laplace method, but that requires $x \to \infty$. Is there another method that works for $x \to 0$?

Martin Sleziak
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p3ngu1n
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  • What is the issue in writing $e^{-xt}$ as its Maclaurin series, then performing termwise integration? – Jack D'Aurizio Nov 11 '18 at 23:48
  • I attempted that but the result I got didn’t seem to make much sense (I couldn’t find a pattern) – p3ngu1n Nov 11 '18 at 23:56
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    There is no pattern to find. Since $e^{-xt}=1-xt+O(x^2 t^2)$, the given integral behaves like $\frac{\pi}{4}-\frac{\pi x}{2}\log(2)+O(x^2)$ as $x\to 0^+$. – Jack D'Aurizio Nov 12 '18 at 00:06

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