Can You Find the Convergent Value of α for this Improper Integral?

[wolski888]

EDIT: I am sorry if you can't understand the integral straight away, I am not familiar with using the notion provided by this forum. I tried, but...

Homework Statement

Find all values of the constant α for which the integral:
∫ [(x/(x^2 + 1)) - (3a/(3x + 1))] dx (from 0 to +infinitity)
converges. Evaluate the integral for these values of α (as a function of α).

The Attempt at a Solution

I don't really know how to approach this. It seems simpler to separate it into 2 separate integrals. And in the integral with the constant 'a', I can sub u = 3x + 1, then du = 3dx
Making it: adu/x

I don't really know if I am on the right track.

Thanks for reading this!
Mike

 

[Dick]

You probably mean adu/u for your substitution. You are started on the right track. Work out both integrals and get the antiderivative as a function of x. Once you get that start working on what the limit of it is as x->infinity. Using the rules of logs will help.

 

[wolski888]

Yes, adu/u. My fault. And thanks!

 

[wolski888]

OK, now I am confused. The ln|3x+1| as x goes to infinity is infinity. So it diverges. But how do I make it converge? Same with the first part of the integral...

 

[Dick]

OK, now I am confused. The ln|3x+1| as x goes to infinity is infinity. So it diverges. But how do I make it converge? Same with the first part of the integral...

Combine the two integral expressions. Use rules of logs. For a special value of a the divergences might cancel.