# Convergence of an integral

• April 13th 2010, 05:03 AM
Gok2
Convergence of an integral
Hey people.
Anyone have any idea why the integral $\int_2^{\infty} \frac{\ln x}{(x-1)\sqrt{x+1}}$ 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!
• April 13th 2010, 05:38 AM
Laurent
Quote:

Originally Posted by Gok2
Hey people.
Anyone have any idea why the integral $\int_2^{\infty} \frac{\ln x}{(x-1)\sqrt{x+1}}$ 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 $0<\epsilon<\frac{1}{2}$. You can note that $\ln x\leq x^{\epsilon}$ when $x$ is large enough (because the ratio goes to 0), hence for such large $x$, the integrand is less that $\frac{x^\epsilon}{(x-1)\sqrt{x+1}}\sim \frac{1}{x^{1+\frac{1}{2}-\epsilon}}$, and the exponent is greater than $1$ because of the choice of epsilon small enough. Hence the convergence.
• April 13th 2010, 06:06 AM
Gok2
Hmm I see, think I got it. thanks a lot!