Find $\displaystyle\int{{{x}^{2}}\cos nx\,\mathrm{d}x}$ where $n$ is a positive integer. He

Question

Find $\displaystyle\int{{{x}^{2}}\cos nx\,\mathrm{d}x}$ where $n$ is a positive integer. Hence find $\displaystyle\int\nolimits_{\pi }^{2\pi }{{{x}^{2}}\cos nx\,\mathrm{d}x}$, giving your answers in the form $a\frac{\pi }{{{n}^{2}}}$, where the possible values of $a$ are to be determined.

Accepted Answer

(i) $\frac{{{x}^{2}}}{n}\sin \left( nx \right)+\frac{2x}{{{n}^{2}}}\cos \left( nx \right)-\frac{2}{{{n}^{3}}}\sin \left( nx \right)+C$; (ii) $\frac{6\pi }{{{n}^{2}}}$ or $\frac{2\pi }{{{n}^{2}}}$; $a=6$ or $2$

N2016 P2 Q2

N2016 P2 Q2