Question 1 - A-Level Maths

Exam Board	OCR MEI
Module	FP2 (Further Pure Mathematics 2)
Year	2012
Session	June
Topic	Integration using inverse trig and hyperbolic functions

1. Differentiate the equation $\sin y = x$ with respect to $x$, and hence show that the derivative of $\arcsin x$ is $\frac { 1 } { \sqrt { 1 - x ^ { 2 } } }$.
2. Evaluate the following integrals, giving your answers in exact form.
  (A) $\int _ { - 1 } ^ { 1 } \frac { 1 } { \sqrt { 2 - x ^ { 2 } } } \mathrm {~d} x$
  (B) $\int _ { - \frac { 1 } { 2 } } ^ { \frac { 1 } { 2 } } \frac { 1 } { \sqrt { 1 - 2 x ^ { 2 } } } \mathrm {~d} x$
A curve has polar equation $r = \tan \theta , 0 \leqslant \theta < \frac { 1 } { 2 } \pi$. The points on the curve have cartesian coordinates $( x , y )$. A sketch of the curve is given in Fig. 1. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{99f0c663-bb5b-4456-854c-df177f5d8349-2_493_796_1123_605} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure} Show that $x = \sin \theta$ and that $r ^ { 2 } = \frac { x ^ { 2 } } { 1 - x ^ { 2 } }$.
Hence show that the cartesian equation of the curve is $$y = \frac { x ^ { 2 } } { \sqrt { 1 - x ^ { 2 } } } .$$ Give the cartesian equation of the asymptote of the curve.