Question

Solve:${\int }_0^{\frac{\pi }{2}}{x}^2{\cos }^{2}xdx$

Answer 1

${\int }_0^{\frac{\pi }{2}}{x}^2{cos}^{2}xdxcomputingtheindefiniteintegral,\int x^2{\cos }^{2}xdx=\frac{1}{3}{x}^3{\cos }^{2}x-\int -\frac{1}{3}{x}^3\sin \left(2x\right)dx\left[integratingbyparts\right]=\frac{1}{3}{x}^3{\cos }^{2}x+\frac{1}{3}(-\frac{1}{2}{x}^3\cos \left(2x\right)+\frac{3}{2}(\frac{1}{2}{x}^2\sin \left(2x\right)+\frac{1}{2}x\cos \left(2x\right)-\frac{1}{4}\sin \left(2x\right)))+CNow,computingtheboundaries,weget{\int }_0^{\frac{\pi }{2}}{x}^2{\cos }^{2}dx=\frac{{\pi }^3-6\pi }{48}-0=\frac{{\pi }^3-6\pi }{48}+C$