Homework Help: Integral proofing

1. Nov 3, 2012

cummings12332

1. The problem statement, all variables and given/known data

let f(x,t)=xe^(-xt).show that the integral I(x)=∫f(x,t)dt (integration from 0 to infinite)exists for all x>=0 . is x->I(x) continuous on [0,infinite)

3. The attempt at a solution
what should i use here to prove the integral exist ???once i prove that exist, can i use the specific integration to see its continuity?????

2. Nov 3, 2012

jbunniii

For a fixed value of $x$, are you able to show that the function $g:\mathbb{R}\rightarrow\mathbb{R}$ defined by $g(t) = xe^{-xt}$ is continuous for all $t$? If so, then it can be integrated on any finite interval, so $\int_{0}^{T} g(t) dt$ exists and is finite for all $T > 0$. You should be able to evaluate this integral explicitly to get some function $G(x,T)$. Then check whether
$$\lim_{T \rightarrow 0}G(x,T)$$
exists.

3. Nov 3, 2012

cummings12332

yes, i have solved it out the limit is 1 , then means I(x)>0 but i feel confused about the continuous part,what should i do in the secound part?

4. Nov 3, 2012

Dick

Ok, so I(x)=1 if x>0. What's I(0)?

5. Nov 4, 2012

cummings12332

ok, i get it , thanks so much