# How do you find the Improper integral #e^[(-x^2)/2]# from #x=-∞# to #x=∞#?

Changing to polar coordinates

To cover the whole plane in polar coordinates we have

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The improper integral of ( e^{-\frac{x^2}{2}} ) from ( x = -\infty ) to ( x = \infty ) cannot be evaluated using elementary functions. This integral represents the area under the standard normal distribution curve, and it is a well-known result in calculus that it cannot be expressed in terms of elementary functions. However, it is equal to ( \sqrt{2\pi} ), which is often derived using advanced techniques such as contour integration in complex analysis.

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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.

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.

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.

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