2011 TPJC P1 Q9

Use the substituition \(u={{\mathrm{e}}^{x}}+2\) to find the exact value of \(\displaystyle\displaystyle \displaystyle\int_{0}^{\ln 2}{\frac{{{\mathrm{e}}^{3x}}}{{{\mathrm{e}}^{x}}+2}\mathrm{d}x}\).[4]

Sketch the graph of \(y={{\mathrm{e}}^{x}}-2\) for \(0\le x\le 2\). State the exact value of \(x\) where the graph crosses the \(x\)-axis.[2]

Hence, find the exact value of \(\ln \,k\) such that

\(\displaystyle\displaystyle \displaystyle\int_{0}^{2}{\left| {{\mathrm{e}}^{x}}-2 \right|\mathrm{d}x}=\displaystyle\displaystyle \displaystyle\int_{e}^{{{e}^{\sqrt{k}}}}{\frac{1}{x\ln x}}\,\mathrm{d}x\).[5]