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