If $\int _1^{\infty }f\left(x\right)dx$ converges absolutely then $\int _1^{\infty }\sin \left(x\right)f\left(x\right)dx$ exists

Question

I'm in need of some assistance with a homework question (I'm doing some calculus work by myself and have gotten stuck on this question):

"Prove or give a counter-example of of the following statement:

Let there be a continuous function $f :[0,\infty) \rightarrow \mathbb{R}.$

If $\int _1^{\infty }f\left(x\right)dx$ converges absolutely then $\int _1^{\infty }\sin \left(x\right)f\left(x\right)dx$ exists"

OK so I know how to disprove the statement if it would have been prove if exists then exists. (Using $\frac{\sin \left(x\right)}{x}$ ) but I can't seem to make any progress for what they actual ask regarding converges absolutely.

Any help is appreciated. Thx!

Answer 1

Note that $\int_1^\infty |f(x)sin(x)|dx\leq \int_1^\infty |f(x)|dx<\infty$, hence the integral $\int_1^\infty f(x)sin(x) dx$ converges absolutely and exists.