integration of exponential functions

Snarf

I have a test tomorrow and I am hoping someone can help me with the following two integrals:

1. [tex]\int[/tex]ye^[tex]^{-y^{2}}[/tex]/2dy

For this one I am using substition with u=-y[tex]^{2}[/tex]/2 and du = -y/4

What I don't understand is how we account for the first y in this integral.


2. [tex]\int[/tex](1/[tex]{\sqrt{2}[/tex])e^[tex]^{-1/6(y-1)^{2}}[/tex]dy

If I'm having trouble with the first one, you can assume I'm not getting far on this one either.

rocomath

you're problem is unclear.

can you re-type that plz.

Snarf

sure. one moment...

neutrino

That is not right. Check the derivative of the y² and then find du/dy.


This one's a bit unclear. Is it [tex]\exp\left(\frac{-1}{6}(y-1)^2\right)[/tex] or [tex]\exp\left(\frac{-1}{6(y-1)^2}\right)[/tex]? Also should there be another y term in the integral somewhere?

JasonRox

The first one is probably a lot easier than you think.

When I see integrals like these, I always solve the derivative of a the exponential part and see what it looks like. What's the derivative of e^(-y^2)?

Snarf

The first one [tex]\exp\left(\frac{-1}{6}(y-1)^2\right)[/tex]

Snarf

-2ye^(-y^2) ?

neutrino

I don't think it's possible to solve the second integral in terms of elementary functions.

Snarf

neutrino

the derivative of -y^2/2 would be -y?

Snarf

Would it help if it was being integrated from negative infinity to infinity?

Should have put that up initially

Snarf

So... for the first one would the answer be -2ye^(-y/2)?

neutrino

Ah. There are some ways of dealing with that.
For example, http://www.physicsforums.com/showthread.php?t=71285

Snarf

or is it -2ye^(y^2)?

neutrino

Neither. The derivative of [tex]e^{-\frac{y^2}{2}}[/tex], with respect to y, [tex]-ye^{-\frac{y^2}{2}}[/tex].

Snarf

Oh. Now I see why you gave me that great hint.

Now as far as your hint on the second one... very interesting, but a bit complex for me. The last time I took a calc class was about three years ago.

Does this look like I'm getting close?

[tex]=\int_{0}^{\infty}\int_{0}^{\infty} e^{-{(1/6(y-1)^2+1/6(x-1)^2)}} dxdy[/tex]

neutrino

You could use a sub. like u = (1/6)(y-1) to simplify things.