Does the integral from 4 to infinity of (3 + sinx)/x diverge?

Question

Does the integral from 4 to infinity of (3+sinx)/x diverge?
I need to use a direct comparsion test to figure this out but I dont know what to comapre it to.
any help is much appreciated

Answer 1

It certainly does. Note that 3 + sin x is between 2 and 4, so for all x > 0 we have
(3 + sin x) / x ≥ 2 / x > 0
and we know the integral to ∞ of 2/x diverges.

Answer 2

Separate it into two integrals: 3/x and sinx / x.

In order for the total integral to be divergent at least one of the those individual terms has to be divergent.

Answer 3

Yes it is simple. Compare it to 1/x.
3+sinx >= 3 -1 since since sinx is >= -1 Therefore
(3 + sinx)/x >= 2/x > 1x. Therefore since the integral of 1/x is divergent the integral of the term is divergent.