integrate e^(-(x^2))dx

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Originally Posted by cmizel

integrate e^(-(x^2))dx

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There's no closed-form solution to the integral, though it can be written as an asymptotic series:

First, rewrite the integral into a form where you can do integration by parts:



Now we can use the integration by parts formulation

,

where



and



That gives you



Now you have a closed-form term, plus a new integral. If you employ integration by parts to solve the new integral, you'll get yet another closed-form term and another new integral (this time with higher powers of x). If you keep going, integrating by parts over and over and over, you build up a series of closed-form terms, and end up with an integral whose value is approaching zero. In the limit, as the number of iterations goes to infinity, you end up with: