- (a) Use the substitution \(x = \sec \theta\) to show that
$$\int \sqrt { } \left( x ^ { 2 } - 1 \right) d x$$
can be written as
$$\int \sec \theta \tan ^ { 2 } \theta \mathrm {~d} \theta$$
(3)
(b) Use integration by parts to show that
$$\int \sec \theta \tan ^ { 2 } \theta \mathrm {~d} \theta = \frac { 1 } { 2 } [ \sec \theta \tan \theta - \ln | \sec \theta + \tan \theta | ] + \text { constant. }$$
(c) Evaluate \(\int _ { 0 } ^ { \frac { \pi } { 4 } } \sin x \sqrt { } ( \cos 2 x ) \mathrm { d } x\).