Find \(\int \frac { 4 + \mathrm { e } ^ { x } } { 2 \mathrm { e } ^ { 2 x } } \mathrm {~d} x\).
Without using a calculator, find \(\int _ { 2 } ^ { 10 } \frac { 1 } { 2 x + 5 } \mathrm {~d} x\), giving your answer in the form \(\ln k\).
\includegraphics[max width=\textwidth, alt={}, center]{f85c4010-17b1-441c-ae8a-e77573d1b0c3-3_446_755_580_735}
The diagram shows the curve \(y = \log _ { 10 } ( x + 2 )\) for \(0 \leqslant x \leqslant 6\). The region bounded by the curve and the lines \(x = 0 , x = 6\) and \(y = 0\) is denoted by \(R\). Use the trapezium rule with 2 strips to find an estimate of the area of \(R\), giving your answer correct to 1 decimal place.