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

1 Find the exact value of \(\int _ { - 1 } ^ { 2 } \left( 4 \mathrm { e } ^ { 2 x } - 2 \mathrm { e } ^ { - x } \right) \mathrm { d } x\).

7

1. By first expanding \(\cos ( 2 \theta + \theta )\), show that \(\cos 3 \theta \equiv 4 \cos ^ { 3 } \theta - 3 \cos \theta\).
2. Find the exact value of \(2 \cos ^ { 3 } \left( \frac { 5 } { 18 } \pi \right) - \frac { 3 } { 2 } \cos \left( \frac { 5 } { 18 } \pi \right)\).
3. Find \(\int \left( 12 \cos ^ { 3 } x - 4 \cos ^ { 3 } 3 x \right) \mathrm { d } x\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

4 Given that \(\int _ { a } ^ { a + 14 } \frac { 1 } { 3 x } \mathrm {~d} x = \ln 2\), find the value of the positive constant \(a\).

7

1. Prove that \(4 \sin x \sin \left( x + \frac { 1 } { 6 } \pi \right) \equiv \sqrt { 3 } - \sqrt { 3 } \cos 2 x + \sin 2 x\).
2. Find the exact value of \(\int _ { 0 } ^ { \frac { 5 } { 6 } \pi } 4 \sin x \sin \left( x + \frac { 1 } { 6 } \pi \right) \mathrm { d } x\).
3. Find the smallest positive value of \(y\) satisfying the equation $$4 \sin ( 2 y ) \sin \left( 2 y + \frac { 1 } { 6 } \pi \right) = \sqrt { 3 } .$$ Give your answer in an exact form.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

8 The expression \(\mathrm { f } ( \theta )\) is defined by \(\mathrm { f } ( \theta ) = 12 \sin \theta \cos \theta + 16 \cos ^ { 2 } \theta\).

1. Express \(\mathrm { f } ( \theta )\) in the form \(R \cos ( 2 \theta - \alpha ) + k\), where \(R > 0,0 < \alpha < \frac { 1 } { 2 } \pi\) and \(k\) is a constant. State the values of \(R\) and \(k\), and give the value of \(\alpha\) correct to 4 significant figures.
2. Find the smallest positive value of \(\theta\) satisfying the equation \(\mathrm { f } ( \theta ) = 17\).
3. Find \(\int f ( \theta ) d \theta\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

3

1. Find \(\int _ { 4 } ^ { 10 } \frac { 4 } { 2 x - 5 } \mathrm {~d} x\), giving your answer in the form \(\ln a\), where \(a\) is an integer.
2. Find the exact value of \(\int _ { 4 } ^ { 10 } \mathrm { e } ^ { 2 x - 5 } \mathrm {~d} x\).

6

1. Show that \(\operatorname { cosec } \theta \left( 3 \sin 2 \theta + 4 \sin ^ { 3 } \theta \right) \equiv 4 + 6 \cos \theta - 4 \cos ^ { 2 } \theta\).
2. Solve the equation $$\operatorname { cosec } \theta \left( 3 \sin 2 \theta + 4 \sin ^ { 3 } \theta \right) + 3 = 0$$ for \(- \pi < \theta < 0\).
3. Find \(\int \operatorname { cosec } \theta \left( 3 \sin 2 \theta + 4 \sin ^ { 3 } \theta \right) \mathrm { d } \theta\).

3 \includegraphics[max width=\textwidth, alt={}, center]{b104e2a7-06c8-4e2e-a4f9-5095ad56897a-04_652_392_274_872} The diagram shows the curve with equation \(y = 6 \mathrm { e } ^ { - \frac { 1 } { 2 } x }\). The points on the curve with \(x\)-coordinates 0 and 2 are denoted by \(A\) and \(B\) respectively. The shaded region is enclosed by the curve, the line through \(A\) parallel to the \(x\)-axis and the line through \(B\) parallel to the \(y\)-axis.

1. Find the exact gradient of the curve at \(B\).
2. Find the exact area of the shaded region.

5

1. Find the quotient when \(6 x ^ { 3 } - 5 x ^ { 2 } - 24 x - 4\) is divided by ( \(2 x + 1\) ), and show that the remainder is 6 .
2. Hence find $$\int _ { 2 } ^ { 7 } \frac { 6 x ^ { 3 } - 5 x ^ { 2 } - 24 x - 4 } { 2 x + 1 } d x$$ giving your answer in the form \(a + \ln b\), where \(a\) and \(b\) are integers.

7

1. Prove that \(\sin 2 x ( \cot x + 3 \tan x ) \equiv 4 - 2 \cos 2 x\).
2. Hence find the exact value of \(\cot \frac { 1 } { 12 } \pi + 3 \tan \frac { 1 } { 12 } \pi\).
3. \includegraphics[max width=\textwidth, alt={}, center]{b104e2a7-06c8-4e2e-a4f9-5095ad56897a-13_796_789_278_708} The diagram shows the curve with equation \(y = 4 - 2 \cos 2 x\), for \(0 < x < 2 \pi\). At the point \(A\), the gradient of the curve is 4 . The point \(B\) is a minimum point. The \(x\)-coordinates of \(A\) and \(B\) are \(a\) and \(b\) respectively. Show that \(\int _ { a } ^ { b } ( 4 - 2 \cos 2 x ) \mathrm { d } x = 3 \pi + 1\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

3 The function f is defined by \(\mathrm { f } ( x ) = \tan ^ { 2 } \left( \frac { 1 } { 2 } x \right)\) for \(0 \leqslant x < \pi\).

1. Find the exact value of \(\mathrm { f } ^ { \prime } \left( \frac { 2 } { 3 } \pi \right)\). \includegraphics[max width=\textwidth, alt={}, center]{dcc483e9-630e-4f02-ad8c-4a27c0720fc6-05_2726_33_97_22}
2. Find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } ( \mathrm { f } ( x ) + \sin x ) \mathrm { d } x\).

5 It is given that \(\int _ { a } ^ { a ^ { 3 } } \frac { 10 } { 2 x + 1 } \mathrm {~d} x = 7\), where \(a\) is a constant greater than 1 .

1. Show that \(a = \sqrt [ 3 ] { 0.5 \mathrm { e } ^ { 1.4 } ( 2 a + 1 ) - 0.5 }\). \includegraphics[max width=\textwidth, alt={}, center]{dcc483e9-630e-4f02-ad8c-4a27c0720fc6-08_2718_35_107_2011} \includegraphics[max width=\textwidth, alt={}, center]{dcc483e9-630e-4f02-ad8c-4a27c0720fc6-09_2725_35_99_20}
2. Use an iterative formula, based on the equation in part (a), to find the value of \(a\) correct to 3 significant figures. Use an initial value of 2 and give the result of each iteration to 5 significant figures.

2 Let \(\mathrm { f } ( x ) = 4 \sin ^ { 2 } 3 x\).

1. Find the value of \(\mathrm { f } ^ { \prime } \left( \frac { 1 } { 4 } \pi \right)\).
2. Find \(\int \mathrm { f } ( x ) \mathrm { d } x\). \includegraphics[max width=\textwidth, alt={}, center]{18aea465-b5b0-48f0-970a-e9ede1dc9370-05_2723_35_101_20}

7

1. Show that \(\tan ^ { 2 } x + \cos ^ { 2 } x \equiv \sec ^ { 2 } x + \frac { 1 } { 2 } \cos 2 x - \frac { 1 } { 2 }\) and hence find the exact value of $$\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \left( \tan ^ { 2 } x + \cos ^ { 2 } x \right) \mathrm { d } x$$
2. \includegraphics[max width=\textwidth, alt={}, center]{0af2714b-d3eb-4112-a869-eda5cf266cd8-13_535_771_274_648} The region enclosed by the curve \(y = \tan x + \cos x\) and the lines \(x = 0 , x = \frac { 1 } { 4 } \pi\) and \(y = 0\) is shown in the diagram. Find the exact volume of the solid produced when this region is rotated completely about the \(x\)-axis.

6

1. 1. Show that \(\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \cos 2 x \mathrm {~d} x = \frac { 1 } { 2 }\).
   2. By using an appropriate trigonometrical identity, find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \sin ^ { 2 } x \mathrm {~d} x\).
2. 1. Use the trapezium rule with 2 intervals to estimate the value of \(\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \sec x d x\), giving your answer correct to 2 significant figures.
   2. Determine, by sketching the appropriate part of the graph of \(y = \sec x\), whether the trapezium rule gives an under-estimate or an over-estimate of the true value.

3 \includegraphics[max width=\textwidth, alt={}, center]{a31a4b4e-83a6-47d9-9679-3471b3da1b6e-2_488_664_863_737} The diagram shows the curve \(y = \mathrm { e } ^ { 2 x }\). The shaded region \(R\) is bounded by the curve and by the lines \(x = 0 , y = 0\) and \(x = p\).

1. Find, in terms of \(p\), the area of \(R\).
2. Hence calculate the value of \(p\) for which the area of \(R\) is equal to 5 . Give your answer correct to 2 significant figures.

7

1. By expanding \(\cos ( 2 x + x )\), show that $$\cos 3 x \equiv 4 \cos ^ { 3 } x - 3 \cos x$$
2. Hence, or otherwise, show that $$\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \cos ^ { 3 } x \mathrm {~d} x = \frac { 2 } { 3 }$$

7

1. By expanding \(\sin ( 2 x + x )\) and using double-angle formulae, show that $$\sin 3 x = 3 \sin x - 4 \sin ^ { 3 } x$$
2. Hence show that $$\int _ { 0 } ^ { \frac { 1 } { 3 } \pi } \sin ^ { 3 } x d x = \frac { 5 } { 24 }$$

7

1. Differentiate \(\ln ( 2 x + 3 )\).
2. Hence, or otherwise, show that $$\int _ { - 1 } ^ { 3 } \frac { 1 } { 2 x + 3 } \mathrm {~d} x = \ln 3$$
3. Find the quotient and remainder when \(4 x ^ { 2 } + 8 x\) is divided by \(2 x + 3\).
4. Hence show that $$\int _ { - 1 } ^ { 3 } \frac { 4 x ^ { 2 } + 8 x } { 2 x + 3 } d x = 12 - 3 \ln 3$$

6

1. Express \(\cos ^ { 2 } x\) in terms of \(\cos 2 x\).
2. Hence show that $$\int _ { 0 } ^ { \frac { 1 } { 3 } \pi } \cos ^ { 2 } x \mathrm {~d} x = \frac { 1 } { 6 } \pi + \frac { 1 } { 8 } \sqrt { } 3$$
3. By using an appropriate trigonometrical identity, deduce the exact value of $$\int _ { 0 } ^ { \frac { 1 } { 3 } \pi } \sin ^ { 2 } x \mathrm {~d} x .$$

3 Find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 6 } \pi } ( \cos 2 x + \sin x ) \mathrm { d } x\).

8 The constant \(a\), where \(a > 1\), is such that \(\int _ { 1 } ^ { a } \left( x + \frac { 1 } { x } \right) \mathrm { d } x = 6\).

1. Find an equation satisfied by \(a\), and show that it can be written in the form $$a = \sqrt { } ( 13 - 2 \ln a )$$
2. Verify, by calculation, that the equation \(a = \sqrt { } ( 13 - 2 \ln a )\) has a root between 3 and 3.5.
3. Use the iterative formula $$a _ { n + 1 } = \sqrt { } \left( 13 - 2 \ln a _ { n } \right)$$ with \(a _ { 1 } = 3.2\), to calculate the value of \(a\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

8

1. Find the equation of the tangent to the curve \(y = \ln ( 3 x - 2 )\) at the point where \(x = 1\).
2. 1. Find the value of the constant \(A\) such that $$\frac { 6 x } { 3 x - 2 } \equiv 2 + \frac { A } { 3 x - 2 }$$
   2. Hence show that \(\int _ { 2 } ^ { 6 } \frac { 6 x } { 3 x - 2 } \mathrm {~d} x = 8 + \frac { 8 } { 3 } \ln 2\).

2 Show that \(\int _ { 0 } ^ { 6 } \frac { 1 } { x + 2 } \mathrm {~d} x = 2 \ln 2\).

4

1. Show that \(\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \cos 2 x \mathrm {~d} x = \frac { 1 } { 2 }\).
2. By using an appropriate trigonometrical identity, find the exact value of $$\int _ { \frac { 1 } { 6 } \pi } ^ { \frac { 1 } { 3 } \pi } 3 \tan ^ { 2 } x \mathrm {~d} x$$