What is the antiderivative of # e^(x^2)#?

Answer 1

Scarlett Allison

It is #sqrtpi/2# times a function called the imaginary error function.

The function #e^(x^2)# has an antiderivative that is not finitely expressible using elementary function.

You can look up the series expansions at #x=0# and at #x=oo# here

https://tutor.hix.ai(x%5E2)+dx

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Answer 2

Charles Arocha

The antiderivative of ( e^{x^2} ) with respect to ( x ) does not have a simple elementary function representation. It cannot be expressed using elementary functions such as polynomials, exponentials, logarithms, trigonometric functions, and their inverses. However, it can be represented using the error function, which is a special function commonly used in mathematical analysis. The antiderivative is given by: ( \int e^{x^2} dx = \frac{\sqrt{\pi}}{2} \text{erf}(x) + C ), where ( C ) is the constant of integration.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.