The complex number \(u\) is given by \(u = 8 - 15 \mathrm { i }\). Showing all necessary working, find the two square roots of \(u\). Give answers in the form \(a + \mathrm { i } b\), where the numbers \(a\) and \(b\) are real and exact.
On an Argand diagram, shade the region whose points represent complex numbers satisfying both the inequalities \(| z - 2 - i | \leqslant 2\) and \(0 \leqslant \arg ( z - i ) \leqslant \frac { 1 } { 4 } \pi\). [0pt]
[4]
\(8 \quad\) Let \(\mathrm { f } ( x ) = \frac { 4 x ^ { 2 } + 9 x - 8 } { ( x + 2 ) ( 2 x - 1 ) }\).
Express \(\mathrm { f } ( x )\) in the form \(A + \frac { B } { x + 2 } + \frac { C } { 2 x - 1 }\).
Hence show that \(\int _ { 1 } ^ { 4 } \mathrm { f } ( x ) \mathrm { d } x = 6 + \frac { 1 } { 2 } \ln \left( \frac { 16 } { 7 } \right)\).