The integral \(\int^{\pi /2}_{0}x^2 \cos^2 x \, dx\) evaluates to \(\frac{\pi^3}{48} - \frac{\pi}{8}\). This result can be utilized to compute the integral \(\int^{\pi / 2}_{- \pi /2} u^2 \cos^2 u \, du\) by recognizing that the integrand is even. Therefore, the latter integral is simply twice the value of the former integral, yielding \(\int^{\pi / 2}_{- \pi /2} u^2 \cos^2 u \, du = 2\left(\frac{\pi^3}{48} - \frac{\pi}{8}\right)\).
PREREQUISITESStudents and professionals in mathematics, particularly those studying calculus, as well as educators looking for examples of integral properties and applications.