calculus - if $int_1^{infty}f(x) mathrm dx$ converges, must $int_1^{infty}f(x)sin x mathrm dx$ converge?

I can't use any of the convergence tests I learned because I have no information on $f(x)$, in particular I don't know if it's continuous or positive.

The only thing I could think of was that if $\displaystyle \int_{1}^{\infty}f(x)\ \mathrm dx$ was absolutely convergent, then $|f(x)\sin x| \leq |f(x)|$ would imply by the comparison test that $\displaystyle \int_{1}^{\infty}f(x)\sin x\ \mathrm dx$ converges.

So if I want to find a counter-example I have to pick $f(x)$ so that $\displaystyle \int_{1}^{\infty}f(x)\ \mathrm dx$ conditionally converges, but I can't think of one.

Answer

Consider $f(x)=\sin(x) / x$.