2019 TJC Promo Q5

Find \(\displaystyle\displaystyle \displaystyle\int{\sin x\cos 3x\,\,\mathrm{d}x}\).[2]

Find \(\displaystyle\displaystyle \displaystyle\int{\frac{x-1}{\sqrt{1+2x-{{x}^{2}}}}}\,\mathrm{d}x\). Find the greatest integer value of \(b\) such that \(\displaystyle\displaystyle \displaystyle\int_{0}^{b}{\frac{x-1}{\sqrt{1+2x-{{x}^{2}}}}}\,\mathrm{d}x\) is defined.[5]

Find \(\displaystyle\displaystyle \displaystyle\int{x\cos x\,\,\mathrm{d}x}\). Hence find the exact value of \(\displaystyle\displaystyle \displaystyle\int_{0}^{2\pi }{x\left| \cos x \right|\mathrm{d}x}\).[5]