It is given that \(y=\sinh \left(x^{2}\right)+\cosh \left(x^{2}\right)\).
(a) Use standard results from the list of formulae (MF19) to find the Maclaurin's series for \(y\) in terms of \(x\) up to and including the term in \(x^{4}\).
(b) Deduce the value of \(\frac{\mathrm{d}^{4} y}{\mathrm{~d} x^{4}}\) when \(x=0\).
(c) Use your answer to part (a) to find an approximation to \(\int_{0}^{\frac{1}{2}} y \mathrm{~d} x\), giving your answer as a rational fraction in its lowest terms.