Obtain a formula for \(\displaystyle\int_0^n x^2\mathrm e^{-2x}\,\mathrm dx\) in terms of \(n\), where \(n>0\).[5]
Hence evaluate \(\displaystyle\int_0^\infty x^2\mathrm e^{-2x}\,\mathrm dx\).[1]
[You may assume that \(n\mathrm e^{-2n}\) and \(n^2\mathrm e^{-2n}\to0\) as \(n\to\infty\).]
The region bounded by the curve \(y=\frac{4x}{x^2+1}\), the \(x\)-axis and the lines \(x=0\) and \(x=1\) is rotated through \(2\pi\) radians about the \(x\)-axis. Use the substitution \(x=\tan\theta\) to show that the volume of the solid obtained is given by \(16\pi\displaystyle\int_0^{\pi/4}\sin^2\theta\,\mathrm d\theta\), and evaluate this integral exactly.[6]