Integration with substitution given

3 It is given that \(x = \ln ( 1 - y ) - \ln y\), where \(0 < y < 1\).

1. Show that \(y = \frac { \mathrm { e } ^ { - x } } { 1 + \mathrm { e } ^ { - x } }\).
2. Hence show that \(\int _ { 0 } ^ { 1 } y \mathrm {~d} x = \ln \left( \frac { 2 \mathrm { e } } { \mathrm { e } + 1 } \right)\).

1. (a) Find

$$\int \frac { 5 + x } { \sqrt { 4 - 3 x ^ { 2 } } } \mathrm {~d} x$$ (b) Hence find the exact value of $$\int _ { 0 } ^ { 1 } \frac { 5 + x } { \sqrt { 4 - 3 x ^ { 2 } } } d x$$ giving your answer in the form \(p \pi \sqrt { 3 } + q\), where \(p\) and \(q\) are rational numbers to be found.

1 \begin{figure}[h] \end{figure} Fig. 8 shows the curve \(y = \mathrm { f } ( x )\), where \(\mathrm { f } ( x ) = \frac { 1 } { \mathrm { e } ^ { x } + \mathrm { e } ^ { - x } + 2 }\).

1. Show algebraically that \(\mathrm { f } ( x )\) is an even function, and state how this property relates to the curve \(y = \mathrm { f } ( x )\).
2. Find \(\mathrm { f } ^ { \prime } ( x )\).
3. Show that \(\mathrm { f } ( x ) = \frac { \mathrm { e } ^ { x } } { \left( \mathrm { e } ^ { x } + 1 \right) ^ { 2 } }\).
4. Hence, using the substitution \(u = \mathrm { e } ^ { x } + 1\), or otherwise, find the exact area enclosed by the curve \(y = \mathrm { f } ( x )\), the \(x\)-axis, and the lines \(x = 0\) and \(x = 1\).
5. Show that there is only one point of intersection of the curves \(y = \mathrm { f } ( x )\) and \(y = \frac { 1 } { 4 } \mathrm { e } ^ { x }\), and find its coordinates.

3 Some years ago an island was populated by red squirrels and there were no grey squirrels. Then grey squirrels were introduced. The population \(x\), in thousands, of red squirrels is modelled by the equation $$x = \frac { a } { 1 + k t }$$ where \(t\) is the time in years, and \(a\) and \(k\) are constants. When \(t = 0 , x = 2.5\).

1. Show that \(\frac { \mathrm { d } x } { \mathrm {~d} t } = - \frac { k x ^ { 2 } } { a }\).
2. Given that the initial population of 2.5 thousand red squirrels reduces to 1.6 thousand after one year, calculate \(a\) and \(k\).
3. What is the long-term population of red squirrels predicted by this model? The population \(y\), in thousands, of grey squirrels is modelled by the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} t } = 2 y - y ^ { 2 }$$ When \(t = 0 , y = 1\).
4. Express \(\frac { 1 } { 2 y - y ^ { 2 } }\) in partial fractions.
5. Hence show by integration that \(\ln \left( \frac { y } { 2 y } \right) = 2 t\). Show that \(y = \frac { 2 } { 1 + \mathrm { e } ^ { - 2 t } }\).
6. What is the long-term population of grey squirrels predicted by this model?

4．Given that \(\mathrm { f } ( x ) = \mathrm { e } ^ { x ^ { 3 } - 2 x }\)
（a）find \(\mathrm { f } ^ { \prime } ( x )\) The curves \(C _ { 1 }\) and \(C _ { 2 }\) are defined by the functions g and h respectively，where $$\begin{array} { l l } \mathrm { g } ( x ) = 8 x ^ { 3 } \mathrm { e } ^ { x ^ { 3 } - 2 x } & x \in \mathbb { R } , x > 0 \\ \mathrm {~h} ( x ) = \left( 3 x ^ { 5 } + 4 x \right) \mathrm { e } ^ { x ^ { 3 } - 2 x } & x \in \mathbb { R } , x > 0 \end{array}$$ （b）Find the \(x\) coordinates of the points of intersection of \(C _ { 1 }\) and \(C _ { 2 }\) Given that \(C _ { 1 }\) lies above \(C _ { 2 }\) between these points of intersection，
（c）find the area of the region bounded by the curves between these two points．
Give your answer in the form \(A + B \mathrm { e } ^ { C }\) where \(A , B\) ，and \(C\) are exact real numbers to be found．

3

1. Given that \(y = 4 x ^ { 3 } - 6 x + 1\), find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
   (l mark)
2. Hence find \(\int _ { 2 } ^ { 3 } \frac { 2 x ^ { 2 } - 1 } { 4 x ^ { 3 } - 6 x + 1 } \mathrm {~d} x\), giving your answer in the form \(p \ln q\), where \(p\) and \(q\) are rational numbers.

6. $$\mathrm { f } ( x ) = \frac { x + 2 } { x ^ { 2 } + 9 }$$

1. Show that $$\int \mathrm { f } ( x ) \mathrm { d } x = A \ln \left( x ^ { 2 } + 9 \right) + B \arctan \left( \frac { x } { 3 } \right) + c$$ where \(c\) is an arbitrary constant and \(A\) and \(B\) are constants to be found.
2. Hence show that the mean value of \(\mathrm { f } ( x )\) over the interval \([ 0,3 ]\) is $$\frac { 1 } { 6 } \ln 2 + \frac { 1 } { 18 } \pi$$
3. Use the answer to part (b) to find the mean value, over the interval \([ 0,3 ]\), of $$\mathrm { f } ( x ) + \ln k$$ where \(k\) is a positive constant, giving your answer in the form \(p + \frac { 1 } { 6 } \ln q\), where \(p\) and \(q\) are constants and \(q\) is in terms of \(k\).