2021 EJC Promo Q11

Find \(\displaystyle\displaystyle \displaystyle\int{{{\sec }^{n+1}}3x\tan 3x\,\mathrm{d}x}\). Where \(n\) is a positive integer.[2]

Find the exact value, in terms of \(\pi \), of \(\displaystyle\displaystyle \displaystyle\int_{0}^{1}{\frac{1}{{{\left( 2x+1 \right)}^{2}}+3}}\,\mathrm{d}x\).[3]

Find the exact value, in terms of \(\mathrm{e}\), of \(\displaystyle\displaystyle \displaystyle\int_{0}^{1}{{{x}^{2}}{{\mathrm{e}}^{2x}}}\,\mathrm{d}x\).[4]

Use the substitution \(x=2\sin u\) to find \(\displaystyle\displaystyle \displaystyle\int{\frac{\sqrt{4-{{x}^{2}}}}{{{x}^{2}}}\,\mathrm{d}x}\).[5]