For integrands such as $e^{-x^{2}}$ is it possible to transform into a Taylor series and exchange the summation with the integral similar to the problem sophomore's dream? $$\int e^{-x^{2}}dx=\int \sum_{n=0}^\infty \frac{(-x^{2})^{n}}{n!}dx$$ I have used the same method to evaluate $\int x^{x}dx$ and I have evaluated this integral numerically too getting the exact answer. Could this method be used to evaluate all non-elementary integrals?
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1$\begingroup$ Of relevance: math.stackexchange.com/questions/83721/… $\endgroup$– DeepakCommented Apr 27, 2020 at 5:40
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1$\begingroup$ @Deepak Thanks just what I need $\endgroup$– hwood87Commented Apr 27, 2020 at 5:48
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