The integral \(\int_0^{\infty} \frac{\sin(ax)}{\sqrt{x}} dx\) can be solved using a substitution \(u = x^{1/2}\), transforming the integral into one involving \(\sin(a u^2)\). By extending the integral to the entire real axis, the solution can be derived from the imaginary part of \(e^{i a u^2}\). Utilizing the Gaussian integral result, \(\int e^{-a y^2} dy = \sqrt{\frac{\pi}{a}}\) for \(\text{Re}(a) > 0\), provides the final answer, although care must be taken with limits or Wick rotation for \(\text{Re}(a) = 0\).
PREREQUISITESMathematics students, particularly those studying calculus and complex analysis, as well as researchers dealing with integrals in physics and engineering applications.