1.08c Integrate e^(kx), 1/x, sin(kx), cos(kx)

1 Show that $$\int _ { 1 } ^ { 4 } \frac { 1 } { 2 x + 1 } \mathrm {~d} x = \frac { 1 } { 2 } \ln 3$$

5 Show that \(\int _ { 1 } ^ { 2 } \left( \frac { 1 } { x } - \frac { 4 } { 2 x + 1 } \right) \mathrm { d } x = \ln \frac { 18 } { 25 }\).

5

1. Express \(\cos ^ { 2 } 2 x\) in terms of \(\cos 4 x\).
2. Hence find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 8 } \pi } \cos ^ { 2 } 2 x \mathrm {~d} x\).

8

1. Find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 3 } \pi } \left( \sin 2 x + \sec ^ { 2 } x \right) \mathrm { d } x\).
2. Show that \(\int _ { 1 } ^ { 4 } \left( \frac { 1 } { 2 x } + \frac { 1 } { x + 1 } \right) \mathrm { d } x = \ln 5\).

3 Show that \(\int _ { 0 } ^ { 1 } \left( \mathrm { e } ^ { x } + 1 \right) ^ { 2 } \mathrm {~d} x = \frac { 1 } { 2 } \mathrm { e } ^ { 2 } + 2 \mathrm { e } - \frac { 3 } { 2 }\).

4

1. Find \(\int \mathrm { e } ^ { 1 - 2 x } \mathrm {~d} x\).
2. Express \(\sin ^ { 2 } 3 x\) in terms of \(\cos 6 x\) and hence find \(\int \sin ^ { 2 } 3 x \mathrm {~d} x\).

2 Show that \(\int _ { 2 } ^ { 6 } \frac { 2 } { 4 x + 1 } \mathrm {~d} x = \ln \frac { 5 } { 3 }\).

4

1. Express \(\cos ^ { 2 } x\) in terms of \(\cos 2 x\).
2. Hence show that $$\int _ { 0 } ^ { \frac { 1 } { 6 } \pi } \left( \cos ^ { 2 } x + \sin 2 x \right) \mathrm { d } x = \frac { 1 } { 8 } \sqrt { } 3 + \frac { 1 } { 12 } \pi + \frac { 1 } { 4 }$$

8

1. By differentiating \(\frac { 1 } { \cos \theta }\), show that if \(y = \sec \theta\) then \(\frac { \mathrm { d } y } { \mathrm {~d} \theta } = \tan \theta \sec \theta\).
2. Hence show that $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} \theta ^ { 2 } } = a \sec ^ { 3 } \theta + b \sec \theta$$ giving the values of \(a\) and \(b\).
3. Find the exact value of $$\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \left( 1 + \tan ^ { 2 } \theta - 3 \sec \theta \tan \theta \right) d \theta$$

6

1. Find \(\int 4 \mathrm { e } ^ { - \frac { 1 } { 2 } x } \mathrm {~d} x\).
2. Show that \(\int _ { 1 } ^ { 3 } \frac { 6 } { 3 x - 1 } \mathrm {~d} x = \ln 16\).

1

1. Find \(\int \frac { 2 } { 4 x - 1 } \mathrm {~d} x\).
2. Hence find \(\int _ { 1 } ^ { 7 } \frac { 2 } { 4 x - 1 } \mathrm {~d} x\), expressing your answer in the form \(\ln a\), where \(a\) is an integer.

3

1. Find \(\int 4 \cos ^ { 2 } \left( \frac { 1 } { 2 } \theta \right) \mathrm { d } \theta\).
2. Find the exact value of \(\int _ { - 1 } ^ { 6 } \frac { 1 } { 2 x + 3 } \mathrm {~d} x\).

2

1. Find \(\int _ { 0 } ^ { a } \left( \mathrm { e } ^ { - x } + 6 \mathrm { e } ^ { - 3 x } \right) \mathrm { d } x\), where \(a\) is a positive constant.
2. Deduce the value of \(\int _ { 0 } ^ { \infty } \left( \mathrm { e } ^ { - x } + 6 \mathrm { e } ^ { - 3 x } \right) \mathrm { d } x\).

5 \includegraphics[max width=\textwidth, alt={}, center]{c703565b-8aa8-424b-9684-6592d4effdf8-2_554_689_1354_726} The diagram shows part of the curve $$y = 2 \cos x - \cos 2 x$$ and its maximum point \(M\). The shaded region is bounded by the curve, the axes and the line through \(M\) parallel to the \(y\)-axis.

1. Find the exact value of the \(x\)-coordinate of \(M\).
2. Find the exact value of the area of the shaded region.

5

1. Find \(\int \left( \tan ^ { 2 } x + \sin 2 x \right) \mathrm { d } x\).
2. Find the exact value of \(\int _ { 0 } ^ { 1 } 3 \mathrm { e } ^ { 1 - 2 x } \mathrm {~d} x\).

7

1. Show that the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 3 } \pi } \left( \cos ^ { 2 } x + \frac { 1 } { \cos ^ { 2 } x } \right) \mathrm { d } x\) is \(\frac { 1 } { 6 } \pi + \frac { 9 } { 8 } \sqrt { } 3\).
2. \includegraphics[max width=\textwidth, alt={}, center]{7e100be2-9768-4fcd-b516-c714e53b0665-3_444_495_1523_865} The diagram shows the curve \(y = \cos x + \frac { 1 } { \cos x }\) for \(0 \leqslant x \leqslant \frac { 1 } { 3 } \pi\). The shaded region is bounded by the curve and the lines \(x = 0 , x = \frac { 1 } { 3 } \pi\) and \(y = 0\). Find the exact volume of the solid obtained when the shaded region is rotated completely about the \(x\)-axis.

1 Find the exact value of \(\int _ { - 1 } ^ { 35 } \frac { 3 } { 2 x + 5 } \mathrm {~d} x\), giving the answer in the form \(\ln k\).

5 It is given that \(\int _ { 0 } ^ { a } \left( 3 \mathrm { e } ^ { 3 x } + 5 \mathrm { e } ^ { x } \right) \mathrm { d } x = 100\), where \(a\) is a positive constant.

1. Show that \(a = \frac { 1 } { 3 } \ln \left( 106 - 5 \mathrm { e } ^ { a } \right)\).
2. Use an iterative formula based on the equation in part (i) to find the value of \(a\) correct to 3 decimal places. Give the result of each iteration to 5 decimal places.

3 The definite integral \(I\) is defined by \(I = \int _ { 0 } ^ { 2 } \left( 4 \mathrm { e } ^ { \frac { 1 } { 2 } x } + 3 \right) \mathrm { d } x\).

1. Show that \(I = 8 \mathrm { e } - 2\).
2. Sketch the curve \(y = 4 \mathrm { e } ^ { \frac { 1 } { 2 } x } + 3\) for \(0 \leqslant x \leqslant 2\).
3. State whether an estimate of \(I\) obtained by using the trapezium rule will be more than or less than \(8 \mathrm { e } - 2\). Justify your answer.

6

1. Show that \(\frac { \cos 2 \theta } { 1 + \cos 2 \theta } \equiv 1 - \frac { 1 } { 2 } \sec ^ { 2 } \theta\).
2. Solve the equation \(\frac { \cos 2 \alpha } { 1 + \cos 2 \alpha } = 13 + 5 \tan \alpha\) for \(0 < \alpha < \pi\).
3. Find the exact value of \(\int _ { - \frac { 1 } { 2 } \pi } ^ { \frac { 1 } { 2 } \pi } \frac { \cos x } { 1 + \cos x } \mathrm {~d} x\).

4

1. Find \(\int \frac { 4 + \sin ^ { 2 } \theta } { 1 - \sin ^ { 2 } \theta } \mathrm {~d} \theta\).
2. Given that \(\int _ { 0 } ^ { a } \frac { 2 } { 3 x + 1 } \mathrm {~d} x = \ln 16\), find the value of the positive constant \(a\).

6

1. Find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \sin x ( 4 \sin x + 6 \cos x ) \mathrm { d } x\).
2. Given that \(\int _ { 0 } ^ { a } \frac { 6 } { 3 x + 2 } \mathrm {~d} x = \ln 49\), find the value of the positive constant \(a\).

2 Show that \(\int _ { 1 } ^ { 7 } \frac { 6 } { 2 x + 1 } \mathrm {~d} x = \ln 125\).

6

1. Show that \(\int _ { 1 } ^ { 6 } \frac { 12 } { 3 x + 2 } \mathrm {~d} x = \ln 256\).
2. Find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 6 } \pi } \left( 8 \sin ^ { 2 } x + \tan ^ { 2 } 2 x \right) \mathrm { d } x\), showing all necessary working.

5 It is given that \(\int _ { 0 } ^ { a } \left( 3 x ^ { 2 } + 4 \cos 2 x - \sin x \right) \mathrm { d } x = 2\), where \(a\) is a constant.

1. Show that \(a = \sqrt [ 3 ] { } ( 3 - 2 \sin 2 a - \cos a )\).
2. Using the equation in part (i), show by calculation that \(0.5 < a < 0.75\).
3. Use an iterative formula, based on the equation in part (i), to find the value of \(a\) correct to 3 significant figures. Give the result of each iteration to 5 significant figures.