Integral of e-x2
By James Thimont (P369) on Tuesday,
April 20, 1999 - 10:13 pm :
Does anyone know the technique for integrating òe-x2 dx?
Supposedly it uses techniques "not taught at school" so I'm not
sure how to do it.
James
By Dave Sheridan (Dms22) on Tuesday,
April 20, 1999 - 10:51 pm :
Hm. This depends on what limits you want to integrate
between (there is no algebraic answer so you can only do a definite integral).
I was taught a method for this at school, as part of Probability/Statistics
A-Level. The ideas are quite straightforward.
Let I=ò-¥¥ e-x2 dx. Now square it. To avoid
getting confused, we'll label one integral dy.
I2=ò-¥¥ò-infty¥ e-x2-y2dx dy
Now transform to polar co-ordinates (please tell me if you would like these
explained). r2=x2+y2 and tanq = y/x.
I2=ò0¥ò02pe-r2r dr dq = p
So
. By symmetry, replacing one of the limits by 0 will halve
the result.
However, for any other limit there is no explicit way of calculating it - just
approximating the integral, although this can be done to any degree of
accuracy you want.
Hope this answers your question in part - if not, give some more information
and we'll see what we can do.
Also, how did you come across this function? It normally occurs as either an
example of a difficult integral, or involved in the Normal distribution; the
latter has some very interesting properties. This is a discussion after all,
so where would you like it to head?
-Dave
By James Thimont (P369) on Monday,
April 26, 1999 - 08:01 pm :
Thanks Dave. I wanted to know if there was an algebraic
method; to evaluate the indefinite integral.
James
By Dave Sheridan (Dms22) on Monday,
April 26, 1999 - 09:25 pm :
...in which case I hope you didn't mind
my long talk about the (rather restricted) definite one.
Just to re-emphasise the point, there is no nice way to express
the indefinite integral; it can just be calculated to good (as
good as you need!) precision. There are a variety of methods to
do this but I don't imagine that's much use to you.
One reason it's not given in school is the notion of converging
approximations isn't particularly developed below degree level;
without it, numerical analysis isn't particularly helpful (try to
answer the question "Why?" and you'll see what I mean...)
Thanks for an interesting question and sorry for there not being
a completely satisfactory answer!
-Dave