5. (a) Given \(0 < a < 1\), sketch the curve with equation $$y = a ^ { x }$$ showing the coordinates of the point at which the curve crosses the \(y\)-axis. The table above shows corresponding values of \(x\) and \(y\) for \(y = x ^ { 2 } + \left( \frac { 1 } { 2 } \right) ^ { x }\) The values of \(y\) are given to 4 significant figures as appropriate.
Using the trapezium rule with all the values of \(y\) in the given table,
(b) obtain an estimate for \(\int _ { 2 } ^ { 4 } \left( x ^ { 2 } + \left( \frac { 1 } { 2 } \right) ^ { x } \right) \mathrm { d } x\) Using your answer to part (b) and making your method clear, estimate
(c) \(\quad \int _ { 2 } ^ { 4 } \left( x ( x - 3 ) + \left( \frac { 1 } { 2 } \right) ^ { x } \right) \mathrm { d } x\)