Integral Proofing: Proving Existence and Continuity of I(x)

[cummings12332]

Homework Statement

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)

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?

 

[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.

 

[cummings12332]

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.

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?

 

[Dick]

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?

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

 

[cummings12332]

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

ok, i get it , thanks so much