NJC Integration Techniques Tutorial Q9

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.

Using the result in part (a), 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.

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