2019 DHS P1 Q10

Find \(\displaystyle\displaystyle \displaystyle\int{\frac{x}{\sqrt{2x-1}}}\,\mathrm{d}x\).[3]

Using the substitution \(t=\tan x\), find \(\displaystyle\displaystyle \displaystyle\int{\frac{1}{4{{\cos }^{2}}x+9{{\sin }^{2}}x}\,\mathrm{d}x}\).[4]

The diagram above shows part of the graph of \(y={{x}^{2}}+3x\) , with rectangles approximating the area under the curve from \(x=0\) to \(x=1\). The area under the curve may be approximated by the total area, \(A\), of \(\left( n-1 \right)\) rectangles each of width \(\frac{1}{n}\). Given that \(\underset{r=1}{\overset{n}{\mathop \sum }}\,{{r}^{2}}=\frac{1}{6}n\left( n+1 \right)\left( 2n+1) \right)\), show that \(A=\frac{\left( n-1 \right)\left( 11n-1 \right)}{6{{n}^{2}}}\).

Explain briefly how the value of \(\mathop{\displaystyle\displaystyle \displaystyle\int }_{0}^{1}{{x}^{2}}+3x\,\mathrm{d}x\) can be deduced from this expression, and hence find this value exactly without integration.[6]