Sorry it's drawn poorly.
Can you explain steps if possible?
https://docs.google.com/drawings/d/1UTwj…
Can you explain steps if possible?
https://docs.google.com/drawings/d/1UTwj…
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Your drawing was perfectly clear and excellent. Seriously. I wish more people would use images of things that are hard to type. I'd much rather look at an image than try to figure out what poorly typed symbols are supposed to mean.
The antiderivative of f(x) = xe^(-x^2) is g(x) = (-1/2)e^(-x^2) + c (we will ignore the c).
This is 0 at -∞ and +∞ so the integral is 0. Gee, that sounds like slim justification for a value of 0, although 0 IS the value. When I plug "the integral from -∞ to +∞ of xdx into a math tool, it says "integral doesn't converge," which makes sense. It's been a LOOONG time since I did any of this stuff. I did a little research and found that this is called an improper integral because of the infinite limits.
I think you need to do this by taking the limit as k-->∞ of the integral from -k to k of f(x)dx.
Note that this is an odd function, i.e, f(x) = -f(-x). So a value of 0 for the integral seems very reasonable, since the integral from -k to k would also be 0 for any k.
One might wonder what the integral from -∞ to 0 is.
That would be lim k_->∞ g(0) - g(-5) = -1/2 - (-1/2)ke^(-k^2) = -1/2. And the integral from 0 to ∞ is 0 -(-1/2) = +1/2.
So forgive a forgetful old man's fuzziness here, but this does seem to cover the necessary bases and give you the correct answer.
The antiderivative of f(x) = xe^(-x^2) is g(x) = (-1/2)e^(-x^2) + c (we will ignore the c).
This is 0 at -∞ and +∞ so the integral is 0. Gee, that sounds like slim justification for a value of 0, although 0 IS the value. When I plug "the integral from -∞ to +∞ of xdx into a math tool, it says "integral doesn't converge," which makes sense. It's been a LOOONG time since I did any of this stuff. I did a little research and found that this is called an improper integral because of the infinite limits.
I think you need to do this by taking the limit as k-->∞ of the integral from -k to k of f(x)dx.
Note that this is an odd function, i.e, f(x) = -f(-x). So a value of 0 for the integral seems very reasonable, since the integral from -k to k would also be 0 for any k.
One might wonder what the integral from -∞ to 0 is.
That would be lim k_->∞ g(0) - g(-5) = -1/2 - (-1/2)ke^(-k^2) = -1/2. And the integral from 0 to ∞ is 0 -(-1/2) = +1/2.
So forgive a forgetful old man's fuzziness here, but this does seem to cover the necessary bases and give you the correct answer.