2017 AJC J2 CT1 P1 Q5

Using integration by parts, show that \(\displaystyle\displaystyle \displaystyle\int_{{{e}^{-1}}}^{1}{{{\left( \ln x \right)}^{2}}}\mathrm{d}x=2-\frac{5}{e}\).[4]

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

The above diagram shows two curves with equations \(y=\ln x\) and \(y=-2\sqrt{\frac{x-1}{x+2}}\).

The line \(y=-1\) intersects the curve \(y=-2\sqrt{\frac{x-1}{x+2}}\) at the point \(A\) where \(x=2\). Use a non-calculator method to find the volume of revolution when the shaded region is rotated completely about the \(x\)-axis.[4]