Find the range of value of $\alpha $ for which $\int_{0}^{\infty}\frac{\sin x}{x^{\alpha}}$ is convergent.
My attempts: $$\sin x \le 1\implies\int_{0}^{\infty}\frac{\sin x}{x^{\alpha}}\le \int_{0}^{\infty}\frac{1}{x^{\alpha}}$$
So $\int_{0}^{\infty}\frac{\sin x}{x^{\alpha}}$ is convergent if $\alpha > 0$.