 Dirichlet test for improper integrals

If $F(t)={\int }_a^{t}f(x)dx$ is bounded and $g$ is a nonnegative differentiable decreasing function tending to zero, then ${\int }_a^{\infty }f(x)g(x)dx$ converges as an improper integral. Integration by parts bounds the tail ${\int }_A^{B}fg$ by a constant multiple of $g(A)$, because $F$ is bounded and ${\int }_A^B|g^{\prime }|=g(A)-g(B)$. In particular ${\int }_1^{\infty }{x}^{-\alpha }\sin xdx$ converges for every $\alpha >0$, even though absolute integrability of that tail requires $\alpha >1$.