(a)[4]
If \(\int_{-2a}^{a} \left( \frac{1}{2}e^{2x} + \frac{1}{4}e^{-x} \right) \, dx = 5\), where \(a\) is a positive constant, show that \(a = \frac{1}{2}\ln\left(10 + \frac{1}{2}e^{-a} + \frac{1}{2}e^{-4a}\right)\).
(b)[2]
Hence, by calculation, show that the value of \(a\) lies between \(1.0\) and \(1.2\).
(c)[3]
Use the iterative formula \(a_{n+1} = \frac{1}{2}\ln\left(10 + \frac{1}{2}e^{-a_n} + \frac{1}{2}e^{-4a_n}\right)\) to determine the value of \(a\) correct to \(4\) significant figures. Record the result of each iteration to \(6\) significant figures.