1.08d Evaluate definite integrals: between limits

10. \begin{figure}[h] \end{figure} Figure 4 shows a sketch of part of the curve with equation $$y = ( 1 + 2 \cos 2 x ) ^ { 2 }$$

1. Show that $$( 1 + 2 \cos 2 x ) ^ { 2 } \equiv p + q \cos 2 x + r \cos 4 x$$ where \(p , q\) and \(r\) are constants to be found. The curve touches the positive \(x\)-axis for the second time when \(x = a\), as shown in Figure 4. The regions bounded by the curve, the \(y\)-axis and the \(x\)-axis up to \(x = a\) are shown shaded in Figure 4.
2. Find, using algebraic integration and making your method clear, the exact total area of the shaded regions. Write your answer in simplest form. \includegraphics[max width=\textwidth, alt={}, center]{9b0b8db0-79fd-4ad5-88c9-737447d9f894-32_2255_51_313_1980}

1. In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.

$$f ( x ) = \frac { 2 x ^ { 3 } - 4 x - 15 } { x ^ { 2 } + 3 x + 4 }$$

1. Show that $$f ( x ) \equiv A x + B + \frac { C ( 2 x + 3 ) } { x ^ { 2 } + 3 x + 4 }$$ where \(A , B\) and \(C\) are integers to be found.
2. Hence, find $$\int _ { 3 } ^ { 5 } \mathrm { f } ( x ) \mathrm { d } x$$ giving your answer in the form \(p + \ln q\), where \(p\) and \(q\) are integers.

1. (a) Using the identity for \(\cos ( A + B )\), prove that

$$\cos 2 A \equiv 2 \cos ^ { 2 } A - 1$$ (b) Hence, using algebraic integration, find the exact value of $$\int _ { \frac { \pi } { 12 } } ^ { \frac { \pi } { 8 } } \left( 5 - 4 \cos ^ { 2 } 3 x \right) d x$$

4. (i) Find $$\int _ { 5 } ^ { 13 } \frac { 1 } { ( 2 x - 1 ) } \mathrm { d } x$$ writing your answer in its simplest form.
(ii) Use integration to find the exact value of $$\int _ { 0 } ^ { \frac { \pi } { 2 } } \sin 2 x + \sec \frac { 1 } { 3 } x \tan \frac { 1 } { 3 } x \mathrm {~d} x$$

3. Given that $$4 x ^ { 3 } + 2 x ^ { 2 } + 17 x + 8 \equiv ( A x + B ) \left( x ^ { 2 } + 4 \right) + C x + D$$

1. find the values of the constants \(A , B , C\) and \(D\).
2. Hence find $$\int _ { 1 } ^ { 4 } \frac { 4 x ^ { 3 } + 2 x ^ { 2 } + 17 x + 8 } { x ^ { 2 } + 4 } d x$$ giving your answer in the form \(p + \ln q\), where \(p\) and \(q\) are integers.

9. \begin{figure}[h] \end{figure}

1. By using the substitution \(u = 2 x + 3\), show that $$\int _ { 0 } ^ { 12 } \frac { x } { ( 2 x + 3 ) ^ { 2 } } \mathrm {~d} x = \frac { 1 } { 2 } \ln 3 - \frac { 2 } { 9 }$$ The curve \(C\) has equation $$y = \frac { 9 \sqrt { x } } { ( 2 x + 3 ) } , \quad x > 0$$ The finite region \(R\), shown shaded in Figure 3, is bounded by the curve \(C\), the \(x\)-axis and the line with equation \(x = 12\). The region \(R\) is rotated through \(2 \pi\) radians about the \(x\)-axis to form a solid of revolution.
2. Use the result of part (a) to find the exact value of the volume of the solid generated.

9. \begin{figure}[h] \end{figure} Diagram not drawn to scale

1. Find $$\int \frac { 1 } { ( 2 x - 1 ) ^ { 2 } } d x$$ Figure 2 shows a sketch of the curve with equation \(y = \mathrm { f } ( x )\) where $$f ( x ) = \frac { 12 } { ( 2 x - 1 ) } \quad 1 \leqslant x \leqslant 5$$ The finite region \(R\), shown shaded in Figure 2, is bounded by the line with equation \(x = 1\), the curve with equation \(y = \mathrm { f } ( x )\) and the line with equation \(y = \frac { 4 } { 3 }\). The region \(R\) is rotated through \(2 \pi\) radians about the \(x\)-axis to form a solid of revolution.
2. Find the exact value of the volume of the solid generated, giving your answer in its simplest form.
   \section*{Leave
   k}

9. (a) Using the formula for \(\sin ( A + B )\) and the relevant double angle formulae, find an
identity for \(\sin 3 x\), giving your answer in the form $$\sin ( 3 x ) \equiv P \sin x + Q \sin ^ { 3 } x$$ where \(P\) and \(Q\) are constants to be determined.
(b) Hence, showing each step of your working, evaluate $$\int _ { \frac { \pi } { 6 } } ^ { \frac { \pi } { 2 } } \sin 3 x \cos x d x$$ (Solutions based entirely on graphical or numerical methods are not acceptable.)

4. $$\mathrm { f } ( x ) = \frac { 4 - 4 x } { x ( x - 2 ) ^ { 2 } } \quad x > 2$$

1. Express \(\mathrm { f } ( x )\) in partial fractions.
2. Hence find \(\int \mathrm { f } ( x ) \mathrm { d } x\)
3. Find $$\int _ { 3 } ^ { 5 } f ( x ) d x$$ giving your answer in the form \(a + \ln b\), where \(a\) and \(b\) are rational numbers to be found.

5. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of the curve with parametric equations $$x = \sqrt { 9 - 4 t } \quad y = \frac { t ^ { 3 } } { \sqrt { 9 + 4 t } } \quad 0 \leqslant t \leqslant \frac { 9 } { 4 }$$ The curve touches the \(x\)-axis when \(t = 0\) and meets the \(y\)-axis when \(t = \frac { 9 } { 4 }\) The region \(R\), shown shaded in Figure 2, is bounded by the curve, the \(x\)-axis and the \(y\)-axis.

1. Show that the area of \(R\) is given by $$K \int _ { 0 } ^ { \frac { 9 } { 4 } } \frac { t ^ { 3 } } { \sqrt { 81 - 16 t ^ { 2 } } } \mathrm {~d} t$$ where \(K\) is a constant to be found.
2. Using the substitution \(u = 81 - 16 t ^ { 2 }\), or otherwise, find the exact area of \(R\).
   (Solutions relying on calculator technology are not acceptable.)

8. \begin{figure}[h] \end{figure} \section*{In this question you must show all stages of your working.} \section*{Solutions relying entirely on calculator technology are not acceptable.} A curve \(C\) has parametric equations $$x = \sin ^ { 2 } t \quad y = 2 \tan t \quad 0 \leqslant t < \frac { \pi } { 2 }$$ The point \(P\) with parameter \(t = \frac { \pi } { 4 }\) lies on \(C\).
The line \(l\) is the normal to \(C\) at \(P\), as shown in Figure 3.

1. Show, using calculus, that an equation for \(l\) is $$8 y + 2 x = 17$$ The region \(S\), shown shaded in Figure 3, is bounded by \(C , l\) and the \(x\)-axis.
2. Find, using calculus, the exact area of \(S\).

1. Given that

$$\frac { 3 x + 4 } { ( x - 2 ) ( 2 x + 1 ) ^ { 2 } } \equiv \frac { A } { x - 2 } + \frac { B } { 2 x + 1 } + \frac { C } { ( 2 x + 1 ) ^ { 2 } }$$

1. find the values of the constants \(A , B\) and \(C\).
2. Hence find the exact value of $$\int _ { 7 } ^ { 12 } \frac { 3 x + 4 } { ( x - 2 ) ( 2 x + 1 ) ^ { 2 } } \mathrm {~d} x$$ giving your answer in the form \(p \ln q + r\) where \(p\), \(q\) and \(r\) are rational numbers.

8. \begin{figure}[h] \end{figure} Figure 3 shows the curve \(C\) with parametric equations $$x = 8 \cos t , \quad y = 4 \sin 2 t , \quad 0 \leqslant t \leqslant \frac { \pi } { 2 } .$$ The point \(P\) lies on \(C\) and has coordinates \(( 4,2 \sqrt { } 3 )\).

1. Find the value of \(t\) at the point \(P\). The line \(l\) is a normal to \(C\) at \(P\).
2. Show that an equation for \(l\) is \(y = - x \sqrt { 3 } + 6 \sqrt { 3 }\). The finite region \(R\) is enclosed by the curve \(C\), the \(x\)-axis and the line \(x = 4\), as shown shaded in Figure 3.
3. Show that the area of \(R\) is given by the integral \(\int _ { \frac { \pi } { 3 } } ^ { \frac { \pi } { 2 } } 64 \sin ^ { 2 } t \cos t \mathrm {~d} t\).
4. Use this integral to find the area of \(R\), giving your answer in the form \(a + b \sqrt { } 3\), where \(a\) and \(b\) are constants to be determined.

2. \begin{figure}[h] \end{figure} Figure 1 shows the finite region \(R\) bounded by the \(x\)-axis, the \(y\)-axis and the curve with equation \(y = 3 \cos \left( \frac { x } { 3 } \right) , 0 \leqslant x \leqslant \frac { 3 \pi } { 2 }\).
The table shows corresponding values of \(x\) and \(y\) for \(y = 3 \cos \left( \frac { x } { 3 } \right)\).

1. Complete the table above giving the missing value of \(y\) to 5 decimal places.
2. Using the trapezium rule, with all the values of \(y\) from the completed table, find an approximation for the area of \(R\), giving your answer to 3 decimal places.
3. Use integration to find the exact area of \(R\).

8. (a) Using the identity \(\cos 2 \theta = 1 - 2 \sin ^ { 2 } \theta\), find \(\int \sin ^ { 2 } \theta \mathrm {~d} \theta\). \begin{figure}[h] \end{figure} Figure 4 shows part of the curve \(C\) with parametric equations $$x = \tan \theta , \quad y = 2 \sin 2 \theta , \quad 0 \leqslant \theta < \frac { \pi } { 2 }$$ The finite shaded region \(S\) shown in Figure 4 is bounded by \(C\), the line \(x = \frac { 1 } { \sqrt { 3 } }\) and the \(x\)-axis. This shaded region is rotated through \(2 \pi\) radians about the \(x\)-axis to form a solid of revolution.
(b) Show that the volume of the solid of revolution formed is given by the integral $$k \int _ { 0 } ^ { \frac { \pi } { 6 } } \sin ^ { 2 } \theta \mathrm {~d} \theta$$ where \(k\) is a constant.
(c) Hence find the exact value for this volume, giving your answer in the form \(p \pi ^ { 2 } + q \pi \sqrt { } 3\), where \(p\) and \(q\) are constants.

5. \begin{figure}[h] \end{figure} Figure 1 shows part of the curve with equation \(x = 4 t \mathrm { e } ^ { - \frac { 1 } { 3 } t } + 3\). The finite region \(R\) shown shaded in Figure 1 is bounded by the curve, the \(x\)-axis, the \(t\)-axis and the line \(t = 8\).

1. Complete the table with the value of \(x\) corresponding to \(t = 6\), giving your answer to 3 decimal places.

   \(t\)	0	2	4	6	8
   \(x\)	3	7.107	7.218		5.223

2. Use the trapezium rule with all the values of \(x\) in the completed table to obtain an estimate for the area of the region \(R\), giving your answer to 2 decimal places.
3. Use calculus to find the exact value for the area of \(R\).
4. Find the difference between the values obtained in part (b) and part (c), giving your answer to 2 decimal places.

3. \begin{figure}[h] \end{figure} Figure 1 shows the finite region \(R\) bounded by the \(x\)-axis, the \(y\)-axis, the line \(x = \frac { \pi } { 2 }\) and the curve with equation $$y = \sec \left( \frac { 1 } { 2 } x \right) , \quad 0 \leqslant x \leqslant \frac { \pi } { 2 }$$ The table shows corresponding values of \(x\) and \(y\) for \(y = \sec \left( \frac { 1 } { 2 } x \right)\).

1. Complete the table above giving the missing value of \(y\) to 6 decimal places.
2. Using the trapezium rule, with all of the values of \(y\) from the completed table, find an approximation for the area of \(R\), giving your answer to 4 decimal places. Region \(R\) is rotated through \(2 \pi\) radians about the \(x\)-axis.
3. Use calculus to find the exact volume of the solid formed.

1. The rectangular hyperbola \(H\) has equation \(x y = c ^ { 2 }\) where \(c\) is a positive constant.

The point \(P \left( c t , \frac { c } { t } \right)\), where \(t > 0\), lies on \(H\)

1. Use calculus to show that an equation of the normal to \(H\) at \(P\) is $$t ^ { 3 } x - t y = c \left( t ^ { 4 } - 1 \right)$$ The parabola \(C\) has equation \(y ^ { 2 } = 6 x\) The normal to \(H\) at the point with coordinates \(( 8,2 )\) meets \(C\) at the point \(Q\) where \(y > 0\)
2. Determine the exact coordinates of \(Q\) Given that
   • the point \(R\) is the focus of \(C\)
   • the line \(l\) is the directrix of \(C\)
   • the line through \(Q\) and \(R\) meets \(l\) at the point \(S\)
   • determine the exact length of \(Q S\)

6. The curve \(H\) has equation $$x y = a ^ { 2 } \quad x > 0$$ where \(a\) is a positive constant. The line with equation \(y = k x\), where \(k\) is a positive constant, intersects \(H\) at the point \(P\)

1. Use calculus to determine, in terms of \(a\) and \(k\), an equation for the tangent to \(H\) at \(P\) The tangent to \(H\) at \(P\) meets the \(x\)-axis at the point \(A\) and meets the \(y\)-axis at the point \(B\)
2. Determine the coordinates of \(A\) and the coordinates of \(B\), giving your answers in terms of \(a\) and \(k\)
3. Hence show that the area of triangle \(A O B\), where \(O\) is the origin, is independent of \(k\)

8. Given that \(z = e ^ { \mathrm { i } \theta }\)

1. show that \(z ^ { n } + \frac { 1 } { z ^ { n } } = 2 \cos n \theta\) where \(n\) is a positive integer.
2. Show that $$\cos ^ { 6 } \theta = \frac { 1 } { 32 } ( \cos 6 \theta + 6 \cos 4 \theta + 15 \cos 2 \theta + 10 )$$
3. Hence solve the equation $$\cos 6 \theta + 6 \cos 4 \theta + 15 \cos 2 \theta = 0 \quad 0 \leqslant \theta \leqslant \pi$$ Give your answers to 3 significant figures.
4. Use calculus to determine the exact value of $$\int _ { 0 } ^ { \frac { \pi } { 3 } } \left( 32 \cos ^ { 6 } \theta - 4 \cos ^ { 2 } \theta \right) d \theta$$ Solutions relying entirely on calculator technology are not acceptable.

1. (a) Use de Moivre's theorem to show that

$$\sin 5 \theta \equiv 16 \sin ^ { 5 } \theta - 20 \sin ^ { 3 } \theta + 5 \sin \theta$$ (b) Hence determine the five distinct solutions of the equation $$16 x ^ { 5 } - 20 x ^ { 3 } + 5 x + \frac { 1 } { 5 } = 0$$ giving your answers to 3 decimal places.
(c) Use the identity given in part (a) to show that $$\int _ { 0 } ^ { \frac { \pi } { 4 } } \left( 4 \sin ^ { 5 } \theta - 5 \sin ^ { 3 } \theta - 6 \sin \theta \right) \mathrm { d } \theta = a \sqrt { 2 } + b$$ where \(a\) and \(b\) are rational numbers to be determined.

3. $$\mathrm { f } ( x ) = \frac { 1 } { x ( 3 x - 1 ) ^ { 2 } } = \frac { A } { x } + \frac { B } { ( 3 x - 1 ) } + \frac { C } { ( 3 x - 1 ) ^ { 2 } }$$

1. Find the values of the constants \(A , B\) and \(C\)
2. 1. Hence find \(\int \mathrm { f } ( x ) \mathrm { d } x\)
   2. Find \(\int _ { 1 } ^ { 2 } \mathrm { f } ( x ) \mathrm { d } x\), giving your answer in the form \(a + \ln b\), where \(a\) and \(b\) are constants.
      (6)

4. \includegraphics[max width=\textwidth, alt={}, center]{d6befd60-de40-41b6-8ae5-48656dbca40c-3_535_1027_276_577} The diagram above shows a sketch of the curve \(C\) with polar equation $$r = 4 \sin \theta \cos ^ { 2 } \theta , \quad 0 \leq \theta < \frac { \pi } { 2 }$$ The tangent to \(C\) at the point \(P\) is perpendicular to the initial line.

1. Show that \(P\) has polar coordinates \(\left( \frac { 3 } { 2 } , \frac { \pi } { 6 } \right)\). The point \(Q\) on \(C\) has polar coordinates \(\left( \sqrt { 2 } , \frac { \pi } { 4 } \right)\).
   The shaded region \(R\) is bounded by \(O P , O Q\) and \(C\), as shown in the diagram above.
2. Show that the area of \(R\) is given by $$\int _ { \frac { \pi } { 6 } } ^ { \frac { \pi } { 4 } } \left( \sin ^ { 2 } 2 \theta \cos 2 \theta + \frac { 1 } { 2 } - \frac { 1 } { 2 } \cos 4 \theta \right) \mathrm { d } \theta$$
3. Hence, or otherwise, find the area of \(R\), giving your answer in the form \(a + b \pi\), where \(a\) and \(b\) are rational numbers.
   (Total 14 marks)

11. (a) Given that \(z = \cos \theta + \mathrm { i } \sin \theta\), use de Moivre's theorem to show that $$z ^ { n } + \frac { 1 } { z ^ { n } } = 2 \cos n \theta$$ (b) Express \(32 \cos ^ { 6 } \theta\) in the form \(p \cos 6 \theta + q \cos 4 \theta + r \cos 2 \theta + \mathrm { s }\), where \(p , q , r\) and \(s\) are integers.
(c) Hence find the exact value of $$\int _ { 0 } ^ { \frac { \pi } { 3 } } \cos ^ { 6 } \theta \mathrm {~d} \theta$$