Differentiate with respect to \(x\) the equation \(a \tan y = x\) (where \(a\) is a constant), and hence show that the derivative of \(\arctan \frac { x } { a }\) is \(\frac { a } { a ^ { 2 } + x ^ { 2 } }\).
By first expressing \(x ^ { 2 } - 4 x + 8\) in completed square form, evaluate the integral \(\int _ { 0 } ^ { 4 } \frac { 1 } { x ^ { 2 } - 4 x + 8 } \mathrm {~d} x\), giving your answer exactly.
Use integration by parts to find \(\int \arctan x \mathrm {~d} x\).
A curve has polar equation \(r = 2 \cos \theta\), for \(- \frac { 1 } { 2 } \pi \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\). Show, by considering its cartesian equation, that the curve is a circle. State the centre and radius of the circle.
Another circle has radius 2 and its centre, in cartesian coordinates, is ( 0,2 ). Find the polar equation of this circle.