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

Answer 1

#-1/(2x)int(-2x*e^(-x^2))=-1/(2x)*(e^(-x^2)+lambda), lambda in RR#

First We should change the expression into our int. We know that #int(u'e^u) =e^u+lambda, lambda in RR.# #int(e^(-x^2)) =-1/(2x)int(-2x*e^(-x^2))# Now We can integrate it properly. (here, #u=-x^2 <=> u'=-2x#)

#-1/(2x)int(-2x*e^(-x^2))=-1/(2x)*(e^(-x^2)+lambda) #

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Answer 2

Emily Ammerman

The integral of ( e^{-x^2} ) with respect to ( x ) does not have an elementary antiderivative expressed in terms of standard mathematical functions. This integral is a well-known non-elementary function and cannot be expressed using elementary functions such as polynomials, exponentials, trigonometric functions, or their inverses. However, it does have significance in probability theory and statistics, particularly in the context of the Gaussian probability distribution function. The integral is denoted by ( \int e^{-x^2} dx ) and is often represented using the error function, denoted by ( \text{erf}(x) ).

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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.