The original problem is to test the convergence of $\int\_{0}^{\infty} \frac{\sin(x)^2}{x^{5/2}}$. It is easy to prove that $\int\_{1}^{\infty} \frac{\sin(x)^2}{x^{5/2}}$ is convergent, but I have checked in <https://www.integral-calculator.com> that $\int\_{0}^{1} \frac{\sin(x)^2}{x^{5/2}}$ is not. How could I prove it? Thanks in advance.