basic integral refresher

[Judge Dredd]

Hi all,
I've been out of math for a few years now and am finding myself stuck on 2 pretty basic integrals. I am pretty confident I'm just forgetting some basic properties but I'd really appreciate if you could take a look at where I am getting stuck and point me in the right direction. Also, I'm not sure what tags to use to make my formatting pretty. If somebody could point me to it I'd be happy to edit the post to improve the format.

1) double integral ( C * x^2 * e^(-x(1+y))) dxdy = 1 where both limits are from 0 to infinity. Find C
I chose to integrate in terms of y first and got
( C * x^2 * e^(-x(1+y)))/ (-x)
Plugging in infinity and 0
(-C * x * e^(-x(1+infinity))) + (C * x * e^(-x))
Can I say this first term is just 0 because e^(-infinity) = 0? This is what I did but I'm not very confident about it.
So integrate C * x * e^(-x) in terms of x from 0 to infinity
C * (-x-1) * e^(-x) -> (C * (-infinity -1) * e^(-infinity)) - (C * (-1) * e^(0))
The first term is 0 and the second term is just C so C = 1?

2) function has value (cy)/x if 0 <= y < x <= 1 and the integral of the function over all values again equals 1. Find C.
So I said double integral of (cy)/x where y ranges from 0 to x, and x ranges from y to 1. I think this is wrong because then my second partal integration will reintroduce the first variable. Am I setting the limits wrong in this case?

Thanks for your time,
Tom

 

[tkhunny]

Plugging in infinity

First, never, EVER say that again.

It's not clear to me what your intermediate result is. After the dx portion, you should have \(\displaystyle \int_{0}^{\infty}\frac{2C}{(y+1)^{3}}\;dy\)