Question

Is there any way to integrate:

Answer 1

\[\int\limits\limits_{0}^{\infty} e^{-t^2}dt\]

Answer 2

-1?

Answer 3

Yes, there is a very nice way if you know multivariable calculus. I'd be glad to explain if you do.

Answer 4

Unfortunately. I'm only a high school AP Cal student. So I'm pretty sure I don't know about that.

Answer 5

The function \[e^{-t^2}\] is what is known as a transcendental function, which basically means it doesn't have an antiderivative that can be written using trig functions, logs, exponentials, etc. Thus, you have to solve problems like this using roundabout ways. This example in particular can be solved using multivariable calculus, though in general finding definite integrals of transcendtal functions is very hard, even with the extra techniques available.

Answer 6

double integral...use polar coordinates

Answer 7

The actual proof is just \[\int_{-\infty}^{\infty}e^{-x^2}dx*\int_{-\infty}^{\infty}e^{-y^2}dy=\int_{-\infty}^{\infty}e^{-x^2+y^2}dxdy\] After switching to polar, you then have this equal to \[\int_0^{2\pi} \int_0^{\infty}re^{-r^2}drd\theta=\pi\] Thus, you have \[\left(\int_{-\infty}^{\infty}e^{-x^2}dx\right)^2=\pi\] and so \[\int_{-\infty}^{\infty}e^{-x^2}dx=\sqrt{\pi}\] Since it is symmetric about the y-axis, this leaves you with \[\int_0^{\infty}e^{-x^2}dx=\frac{\sqrt{\pi}}{2}\]

Answer 8

Thanks, I'll look up on this method for the week. My brain just froze. :D

Answer 9

No problem! It's nothing too difficult, it doesn't use any techniques you wouldn't learn in any multivariable calculus course, so it's more just wrapping your head around the concepts. In particular, I'd google/wiki "order of integration" and "Jacobian" if you want to learn about some of the things I used in the proof.

Answer 10

I understand that if it's infinity, we can find it out by diving by two. But if it's a number like -1 to 1. WHat do we do? We can't divide?

Answer 11

good answer imperialist

Answer 12

@hmm, that's another interesting thing about this problem. Even this method only works for finding the integral when the bounds are negative infinity to infinity (or zero to infinity). Otherwise, even this method fails. There may be some other cases where exact answers can be found using solvable methods (though I don't know any, for e^(-t^2) at least). In those cases, the best way to solve it would be by estimation, through any one of a dozen ways (midpoint rule, trapezoid rule, simpson's rule, power series, etc.).