2019 ASRJC P1 Q1

Show that \(\sqrt{\frac{1-2x}{1+2x}}\) can be written in the form \(\frac{\mathrm{f}\left( x \right)}{\sqrt{1-4{{x}^{2}}}}\) where \(\mathrm{f}\left( x \right)\) is a polynomial to be determined. Hence, find \(\displaystyle\displaystyle \displaystyle\int{\sqrt{\frac{1-2x}{1+2x}}\,\mathrm{d}x}\).[3]

Show that \(\displaystyle\displaystyle \displaystyle\int_{\mathrm{e}}^{{{\mathrm{e}}^{2}}}{\frac{1}{x\ln x}}\mathrm{ d}x=\ln 2\).[3]