Given that \(\mathrm { f } ( t ) = \arcsin t\), write down an expression for \(\mathrm { f } ^ { \prime } ( t )\) and show that
$$\mathrm { f } ^ { \prime \prime } ( t ) = \frac { t } { \left( 1 - t ^ { 2 } \right) ^ { \frac { 3 } { 2 } } }$$
Show that the Maclaurin expansion of the function \(\arcsin \left( x + \frac { 1 } { 2 } \right)\) begins
$$\frac { \pi } { 6 } + \frac { 2 } { \sqrt { 3 } } x$$
and find the term in \(x ^ { 2 }\).
Sketch the curve with polar equation \(r = \frac { \pi a } { \pi + \theta }\), where \(a > 0\), for \(0 \leqslant \theta < 2 \pi\).
Find, in terms of \(a\), the area of the region bounded by the part of the curve for which \(0 \leqslant \theta \leqslant \pi\) and the lines \(\theta = 0\) and \(\theta = \pi\).
Find the exact value of the integral
$$\int _ { 0 } ^ { \frac { 3 } { 2 } } \frac { 1 } { 9 + 4 x ^ { 2 } } \mathrm {~d} x$$