- (a) Show that, under the substitution \(x = \frac { 3 } { 4 } \sinh u\),
$$\int \frac { x ^ { 2 } } { \sqrt { 16 x ^ { 2 } + 9 } } \mathrm {~d} x = k \int ( \cosh 2 u - 1 ) \mathrm { d } u$$
where \(k\) is a constant to be determined.
(b) Hence show that
$$\int _ { 0 } ^ { 1 } \frac { 64 x ^ { 2 } } { \sqrt { 16 x ^ { 2 } + 9 } } \mathrm {~d} x = p + q \ln 3$$
where \(p\) and \(q\) are rational numbers to be found.