Evaluating improper intergral with trig function

[SteliosVas]

Homework Statement

Okay so the problem is asking simply for a proof for convergence /divergence of the following indefinite integral:

∫(x*sin²(x))/(x³-1) over [2,∞)

Homework Equations

I know I can use substitution method

The Attempt at a Solution

Okay so I know if i factorize the bottom part of the integral I get:

∫(x*sin²(x))/(x-1)(x-1)(x²+x+1).

Now obviously I am not sure if this a very elemetary problem or indeed quite challenging. I have tried substituion making u = (x³-1) but than didn't really work... Not sure what else I should try

 

[showzen]

I don't think that the problem wants you to solve the improper integral. Rather, it wants you to perform some test for convergence / divergence.

 

[SteliosVas]

I don't think that the problem wants you to solve the improper integral. Rather, it wants you to perform some test for convergence / divergence.

That's what I thought, as the computation of the improper integral would probably be too challenging... Should I use the limit test? In other words compare the overall function to the convergence/divergence of 1/x³ as it approaches 2

 

[Ray Vickson]

That's what I thought, as the computation of the improper integral would probably be too challenging... Should I use the limit test? In other words compare the overall function to the convergence/divergence of 1/x³ as it approaches 2

You don't need to worry about the lower limit x = 2, because nothing bad happens to the integrand $f(x) = x \sin^2(x)/(x^3-1)$ as $x \to 2$. However, you do need to worry about the upper limit as $x \to \infty$. Your textbook and/or course notes must have some relevant tests to use; try out some of them, then come back here if you still have some questions and issues.