This is good revision for me!
Hume:Trusting scientific knowledge is as good as faith in a religion because science is based on inductive reasoning.
1. If we lived in a non-inductive universe and all our knowledge were to be deductively sound, then all possible effects of a particular cause are equally valid. (There is no experience to differentiate between the possibility of a ball falling down or a ball defying gravity and floating upwards, for example).
2. However, we show a preference to the possibility that the ball will indeed fall.
3. So clearly, we live in a universe where our preferences for such possibilities are governed by inductively reasoned laws of nature.
4. Inductive reasoning assumes that the future will resemble the past. (This is called the uniformity principle).
5. The universe is rather chaotic and doesn't really work that way. (Hume doesn't specifically say this, though I think it's implied that he invokes the uniformity principle).
6. So inductive reasoning isn't a rationally sound base for our knowledge at all.
7. Because science is based on induction, there really isn't any rationally sound basis of trusting scientific knowledge.
Note that Hume doesn't discount the usefulness of science. He's just saying that the Problem of Induction entails that scientific laws cannot be the be all and end all of knowledge.
Popper:
Popper accepts the problem of induction. However, from my interpretation of the text, he seems to be attempting to salvage rational grounds of scientific knowledge by applying deduction to all that inductive reasoning.
His solution to the problem of demarcation is falsifiability.
Falsifiability is
modus tollens (If p, then q. Not q, therefore not p.) and is a valid deductive argument.
By attempting to falsify theories and eliminating them as a possible scientific law, Popper hopes to introduce rationality back into scientific knowledge.
Again, this is very modest, just like Hume. He's not saying that falsification will be the saviour to all inductive problems. He's saying is a solution to the problem of demarcation and might possibly give us a reason to trust science again.
Kuhn:
Kuhn's scientific revolutions fundamentally rejects science as teleological. Science isn't truth, and isn't set out to discover truth (as in THE TRUTH). If there is a truth, science may happen upon it, but we will never know.
Normal science is scientific progress - this is the real productive side of science. We solve problems, cure cancer, invent mobile phones and other useful stuff. However, this only offers efficiency and productivity, not actual epistemic truth.
So I think Kuhn would actually agree with Hume in that he "undermines the integrity of scientific knowledge" (as per your question). I don't know if he "renders it as dogmatic as any religion", because he probably doesn't see science as attempting Truth at all. Hume seems to be actively rejecting science as rationally sound (deductive), but in comparison Kuhn really just ignores all this Truth business and talks about how historically science evolves in different paradigms, solving problems along the way.
Hope that helps, hope I got my stuff right. If I got anything wrong let me know!