Actually, I know that the integral diverge. But Im having trouble proving it.
Here's what I've tried:
we know that for each point that the $ sin=0 $ the value of the function is $ 1 $, hence, for each $ x_{k}=\frac{\pi k}{5} $ it follows that $ f(x_k) = 1 $.
since $ f $ is continious, there is a $ \delta_k $ such that for any $ y_{k}\in(x_{k},x_{k}+\delta_{k}) $, it follows that in the interval $ [x_{k},y_{k}] $ , $ f(x) > \frac{1}{2} $.
If I could find an explicit expression for $ y_k $ that is depend on $ k $, I could evaluate the integral, since the inregrand is a positive function, it follows that :
$ \intop_{0}^{\infty}\frac{dx}{1+\left(x\sin5x\right)^{2}}>\sum_{k=1}^{\infty}\intop_{x_{k}}^{y_{k}}f\left(x\right)dx\geq\sum_{k=1}^{\infty}\frac{1}{2}|y_{k}-x_{k}| $
But Im not sure how to evaluate the $ \delta_k $ or the $ y_k $ in order to continue.
Thanks in advance.
