Using the binomial expansion, or otherwise, express \(( 1 - x ) ^ { 3 }\) in ascending powers of \(x\).
Show that the expansion of
$$( 1 + y ) ^ { 4 } - ( 1 - y ) ^ { 3 }$$
is
$$7 y + p y ^ { 2 } + q y ^ { 3 } + y ^ { 4 }$$
where \(p\) and \(q\) are constants to be found.
Hence find \(\int \left[ ( 1 + \sqrt { x } ) ^ { 4 } - ( 1 - \sqrt { x } ) ^ { 3 } \right] \mathrm { d } x\), expressing each coefficient in its simplest form.