How to test if a definite integral is finite or not?

[arroy_0205]

Suppose I have a complicated integral whose exact evaluation seems extremely difficult or may be even impossible, in such a case is there any way to tell if the integration result is finite or not? suppose the problem is
<br /> \int_{-\infty}^{\infty} f(x;a,b) dx<br />
I think there might be some conditions on the function involved. Actually the function contains some parameters also (a,b) which can be taken to be constants for a particular case. Now I am looking for a condition general enough to handle arbitrary parameters, ie, can I tell if the integral is finite for any arbitrary values of those parameters? If yes, then how or under what condition? Take as an example:
<br /> \int_{-\infty}^{\infty} e^{-2a \tanh^2(bx)} dx<br />
Is this finite?

 

[mathman]

I may be a little rusty. If I remember correctly, |tanh(bx)| -> 1 as x -> oo. In which case, the integral certainly is NOT finite. A necessary condition for integral to be finite with an infinite domain is the integrand go to zero as x -> oo. Also, this condition is not sufficient!

 

[Zizy]

If i remember correctly, besides using definition you can use the fact that any positive (or negative) function has a finite integral, if its falling with a greater exponent than 1/x. (equivalently - if its below x^-(1+eps) for any x greater than some x0). It doesn't work ok for an alternating function.
This applies if you are interested in a true integral, not prime value. Also, this covers just what happens when x goes to infinity, not points where function itself diverges.