Does x/(x^2 - 1)^2 converge or diverge?

[IntegrateMe]

Does the integral from 2 to infinity of x/(x^2 - 1)² converge or diverge, if it converges, state its value.

I got up to the point where i know the integral converges using the direct comparison test and comparing the original integral with the integral from 2 to infinity of 1/x³.

First, i "FOILed" the bottom to get:

x/[x⁴ - 2x² + 1]

Using my FOILed integral i took out an x/x⁴ which is = to 1/x³ and then used the direct comparison test to find out that the integral from 2 to infinity of 1/x³ reaches a value of 1/2 (converging), thus allowing us to assume that the original integral converges as well.

Now, i just don't know how to calculate the value of convergence for x/(x^2 - 1)². Any advice to lead me in the right direction?

 

[Count Iblis]

Hint: The numerator is proportional to the derivative of the contents of the brackets in the denominator.

 

[IntegrateMe]

u-sub ;)

 

[Hurkyl]

Don't do it this way^* -- but is there any reason why you did not consider partial fractions?

*: unless you want to.

 

[IntegrateMe]

i just realized you have to use u-substitution. partial fractions take much too long for a problem like this.

 

[Hurkyl]

I bet you could have done it in the half hour between your previous two posts.