How to prove this is a tautology $\forall x : (p(x) \wedge \neg q(x)) \vee \exists x: \neg p(x) \vee \neg \exists x: q(x)$
The problem is I know that this is a tautology, but I don't know how to prove it. Can anyone can help me?

Try breaking it up in words.  The second two conditions are:
"there is an $x$ such that $p(x)$ is false"
"there is no $x$ such that $q(x)$ is true"
You need to show that if neither of these conditions holds (that is, both of these statements are false), then necessarily the first one ("for every $x$, $p(x)$ is true and $q(x)$ is false") must be true.  
Can you see how to prove that?
