N2023 P1 Q4

Find \(\displaystyle\displaystyle \displaystyle\int{\cos px\,\,\cos \,\,qx\,\mathrm{d}x}\), where \(p\) and \(q\) are constants such that \(p\ne q\) and \(p\ne -q\).[2]

Given that \(n\ne 0\), show that \(\displaystyle\displaystyle \displaystyle\int{x\cos nx\,\,\mathrm{d}x}=\frac{x\sin nx}{n}+\frac{\cos nx}{{{n}^{2}}}+c\), where \(c\) is an arbitrary constant.[3]

Using the result in part (b) show that, for all positive integers \(n\), the value of \(\displaystyle\displaystyle \displaystyle\int_{0}^{\pi }{x\cos nx\,\mathrm{d}x}\) can be expressed as \(\frac{k}{{{n}^{2}}}\), where the possible value(s) of \(k\) are to be determined.[2]

Using the result in part (b) find the exact value of \(\displaystyle\displaystyle \displaystyle\int_{0}^{\frac{\pi }{2}}{\left| x\cos 2x \right|\mathrm{d}x}\).[3]