4．（a）Find the binomial series expansion for \(( 4 + y ) ^ { \frac { 1 } { 2 } }\) in ascending powers of \(y\) up to and including the term in \(y ^ { 3 }\) ．Simplify the coefficient of each term．
（3）
（b）Hence show that the binomial series expansion for \(\left( 4 + 5 x + x ^ { 2 } \right) ^ { \frac { 1 } { 2 } }\) in ascending powers of \(x\) up to and including the term in \(x ^ { 3 }\) is $$2 + \frac { 5 x } { 4 } - \frac { 9 x ^ { 2 } } { 64 } + \frac { 45 x ^ { 3 } } { 512 }$$ （c）Show that the binomial series expansion of \(\left( 4 + 5 x + x ^ { 2 } \right) ^ { \frac { 1 } { 2 } }\) will converge for \(- \frac { 1 } { 2 } \leqslant x \leqslant \frac { 1 } { 2 }\)
（d）Use the result in part（b）to estimate $$\int _ { - \frac { 1 } { 2 } } ^ { \frac { 1 } { 2 } } \sqrt { 4 + 5 x + x ^ { 2 } } d x$$ Give your answer as a single fraction．