Choosing constants to make integral convergent

[sr3056]

I want to integrate γβ^γx/(β + x)^γ+1 from 0 to ∞ (given β and γ are both > 0)

So for large x the integrand is approximately proportional to x^-γ

So for which values of γ is the integral defined? Surely for any γ > 0 the integrand tends to zero as x tends to infinity?

Thanks

 

[mathman]

You need γ > 1 to integrate to ∞. The integral of 1/x is divergent for x -> ∞.

 

[sr3056]

But 1/x² etc. is ok?

 

[haruspex]

But 1/x² etc. is ok?

Yes, in fact ∫x^-(1+ε) converges for any ε > 0. Even ∫1/(x ln^1+ε(x)) converges, but not ∫1/(x ln(x)) , nor ∫1/(x ln(x)ln(ln(x))) etc.

 

[sr3056]

Thanks that's really helpful. Is it because the ones that don't exist aren't going to zero fast enough?

And what about non-negative powers - are there any special cases where these can be integrated to infinity that I should be aware of?

e.g. ∫x²

 

[chiro]

Hey sr3056.

For this problem I'm assuming that you are integrating with respect to x and that the other parameters are independent from x.

It might help if you make a change of variable u = β + x, change your limits and then integrate with respect to this.

The reason is that you will get (u-β)/u^[y+1] which can be analytically integrated under which you can get a lower and upper limit and decide exactly what your parameter must be for it to converge.