Integrating w/o U-Substitution

[CeceBear]

Homework Statement

∫y=e^(-2x^2)dx

Homework Equations

The Attempt at a Solution

I can't recall any method for this. I know that the integral of e^x is e^x, but I know that in this case the integral would not be the same as the original function because the derivative of -2x^2 would be -4x. Can anyone help me?

 

[Curious3141]

Short answer: You can't evaluate that integral in the usual way.

Long answer: This is related to the Gaussian integral:

$$\int_{-\infty}^{\infty}e^{-x^2}dx$$

which you should be able to see by making the substitution $u = \sqrt{2}x$ into your integral.

The indefinite integral of the function $e^{-x^2}$ cannot be expressed in terms of elementary functions. There is a related function, called the error function, written as $erf(x)$, which involves a definite integral of the same function between finite bounds $0$ and $x$. Again, the error function cannot be expressed in terms of elementary functions, and is simply defined in terms of that integral.

 

[andrien]

definite integral of it can be evaluated between limits 0 to infinity by transforming to polar coordinates.