Is there really no way to integrate e−x2e^{-x^2}?

Today in my calculus class, we encountered the function ex2, and I was told that it was not integrable.

I was very surprised. Is there really no way to find the integral of ex2? Graphing ex2, it appears as though it should be.

A Wikipedia page on Gaussian Functions states that


This is from -infinity to infinity. If the function can be integrated within these bounds, I’m unsure why it can’t be integrated with respect to (a,b).

Is there really no way to find the integral of ex2, or are the methods to finding it found in branches higher than second semester calculus?


That function is integrable. As a matter of fact, any continuous function (on a compact interval) is Riemann integrable (it doesn’t even actually have to be continuous, but continuity is enough to guarantee integrability on a compact interval). The antiderivative of ex2 (up to a constant factor) is called the error function, and can’t be written in terms of the simple functions you know from calculus, but that is all.

Source : Link , Question Author : Zolani13 , Answer Author : JLA

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