# Thread: Convergence of an integral

1. ## Convergence of an integral

Hey people.
Anyone have any idea why the integral $\displaystyle \int_2^{\infty} \frac{\ln x}{(x-1)\sqrt{x+1}}$ converges ?
I tried to use all the tricks I know, Dirichlet's test for convergence of integrals, the comparison test , but nothing worked for me....
Any clue how to show that this integral converges?

Thanks!

2. Originally Posted by Gok2
Hey people.
Anyone have any idea why the integral $\displaystyle \int_2^{\infty} \frac{\ln x}{(x-1)\sqrt{x+1}}$ converges ?
I tried to use all the tricks I know, Dirichlet's test for convergence of integrals, the comparison test , but nothing worked for me....
Any clue how to show that this integral converges?

Thanks!
Choose $\displaystyle 0<\epsilon<\frac{1}{2}$. You can note that $\displaystyle \ln x\leq x^{\epsilon}$ when $\displaystyle x$ is large enough (because the ratio goes to 0), hence for such large $\displaystyle x$, the integrand is less that $\displaystyle \frac{x^\epsilon}{(x-1)\sqrt{x+1}}\sim \frac{1}{x^{1+\frac{1}{2}-\epsilon}}$, and the exponent is greater than $\displaystyle 1$ because of the choice of epsilon small enough. Hence the convergence.

3. Hmm I see, think I got it. thanks a lot!