2016 VJC Promo Q10

Find \(\displaystyle\displaystyle \displaystyle\int{\frac{2-x}{\sqrt{(4x-{{x}^{2}}-1)}}\,\mathrm{d}x}\).[2]

Find \(\displaystyle\displaystyle \displaystyle\int{\frac{1}{\sqrt{(4x-{{x}^{2}}-1)}}\,\mathrm{d}x}\).[2]

Using the results of part (i) and (ii), find \(\displaystyle\displaystyle \displaystyle\int{\frac{x}{\sqrt{(4x-{{x}^{2}}-1)}}\,\mathrm{d}x}\).[2]

By using the substitution \(u=\sqrt{x}\), find \(\displaystyle\displaystyle \displaystyle\int{\frac{\sqrt{x}}{1+\sqrt{x}}\,\mathrm{d}x}\).[4]

Find the exact value of \(\displaystyle\displaystyle \displaystyle\int_{0}^{9}{\ln \left( 1+\sqrt{x} \right)\,\mathrm{d}x}\), leaving your answer in the form \(a\ln b-\frac{c}{b}\), where \(a\), \(b\) and \(c\) are integers to be determined.[4]