Iterative or numerical methods after integration

7
\includegraphics[max width=\textwidth, alt={}, center]{e2cc23d2-f3ac-488b-97e1-79e2a98a87ba-3_421_885_251_628} The diagram shows part of the curve \(y = \cos ( \sqrt { } x )\) for \(x \geqslant 0\), where \(x\) is in radians. The shaded region between the curve, the axes and the line \(x = p ^ { 2 }\), where \(p > 0\), is denoted by \(R\). The area of \(R\) is equal to 1 .

1. Use the substitution \(x = u ^ { 2 }\) to find \(\int _ { 0 } ^ { p ^ { 2 } } \cos ( \sqrt { } x ) \mathrm { d } x\). Hence show that \(\sin p = \frac { 3 - 2 \cos p } { 2 p }\).
2. Use the iterative formula \(p _ { n + 1 } = \sin ^ { - 1 } \left( \frac { 3 - 2 \cos p _ { n } } { 2 p _ { n } } \right)\), with initial value \(p _ { 1 } = 1\), to find the value of \(p\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

10

1. Express \(\frac { 1 } { ( 3 - x ) ( 6 - x ) }\) in partial fractions.
2. In a chemical reaction, the amount \(x\) grams of a substance at time \(t\) seconds is related to the rate at which \(x\) is changing by the equation $$\frac { \mathrm { d } x } { \mathrm {~d} t } = k ( 3 - x ) ( 6 - x )$$ where \(k\) is a constant. When \(t = 0 , x = 0\) and when \(t = 1 , x = 1\).
   (a) Show that \(k = \frac { 1 } { 3 } \ln \frac { 5 } { 4 }\).
   (b) Find the value of \(x\) when \(t = 2\).

7 In this question you must show detailed reasoning.
Use the substitution \(u = 6 x ^ { 2 } + x\) to solve the equation \(36 x ^ { 4 } + 12 x ^ { 3 } + 7 x ^ { 2 } + x - 2 = 0\).

5 In this question you must show detailed reasoning. Using the substitution \(\mathrm { u } = \mathrm { x } + 1\), find the value of the positive integer \(c\) such that \(\int _ { \mathrm { c } } ^ { \mathrm { c } + 4 } \frac { \mathrm { x } } { ( \mathrm { x } + 1 ) ^ { 2 } } \mathrm { dx } = \ln 3 - \frac { 1 } { 3 }\).