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

9 A curve which passes through \(( 0,3 )\) has equation \(y = \mathrm { f } ( x )\). It is given that \(\mathrm { f } ^ { \prime } ( x ) = 1 - \frac { 2 } { ( x - 1 ) ^ { 3 } }\).

1. Find the equation of the curve.
   The tangent to the curve at \(( 0,3 )\) intersects the curve again at one other point, \(P\).
2. Show that the \(x\)-coordinate of \(P\) satisfies the equation \(( 2 x + 1 ) ( x - 1 ) ^ { 2 } - 1 = 0\).
3. Verify that \(x = \frac { 3 } { 2 }\) satisfies this equation and hence find the \(y\)-coordinate of \(P\).

2 The function f is defined by \(\mathrm { f } ( x ) = \frac { 2 } { ( x + 2 ) ^ { 2 } }\) for \(x > - 2\).

1. Find \(\int _ { 1 } ^ { \infty } \mathrm { f } ( x ) \mathrm { d } x\).
2. The equation of a curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \mathrm { f } ( x )\). It is given that the point \(( - 1 , - 1 )\) lies on the curve. Find the equation of the curve.

10 \includegraphics[max width=\textwidth, alt={}, center]{616a6177-0d5c-49f7-b0c1-9138a13c1963-4_687_488_262_826} The diagram shows the part of the curve \(y = \frac { 8 } { x } + 2 x\) for \(x > 0\), and the minimum point \(M\).

1. Find expressions for \(\frac { \mathrm { d } y } { \mathrm {~d} x } , \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) and \(\int y ^ { 2 } \mathrm {~d} x\).
2. Find the coordinates of \(M\) and determine the coordinates and nature of the stationary point on the part of the curve for which \(x < 0\).
3. Find the volume obtained when the region bounded by the curve, the \(x\)-axis and the lines \(x = 1\) and \(x = 2\) is rotated through \(360 ^ { \circ }\) about the \(x\)-axis.

9 A curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \sqrt { } ( 4 x + 1 )\) and \(( 2,5 )\) is a point on the curve.

1. Find the equation of the curve.
2. A point \(P\) moves along the curve in such a way that the \(y\)-coordinate is increasing at a constant rate of 0.06 units per second. Find the rate of change of the \(x\)-coordinate when \(P\) passes through \(( 2,5 )\).
3. Show that \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } \times \frac { \mathrm { d } y } { \mathrm {~d} x }\) is constant.

4 A curve with equation \(y = \mathrm { f } ( x )\) passes through the point \(A ( 3,1 )\) and crosses the \(y\)-axis at \(B\). It is given that \(\mathrm { f } ^ { \prime } ( x ) = ( 3 x - 1 ) ^ { - \frac { 1 } { 3 } }\). Find the \(y\)-coordinate of \(B\).

2 A curve has equation \(y = f ( x )\). It is given that \(f ^ { \prime } ( x ) = \frac { 1 } { \sqrt { } ( x + 6 ) } + \frac { 6 } { x ^ { 2 } }\) and that \(f ( 3 ) = 1\). Find \(f ( x )\).

6

1. Prove that $$\sin 2 \theta ( \operatorname { cosec } \theta - \sec \theta ) \equiv \sqrt { 8 } \cos \left( \theta + \frac { 1 } { 4 } \pi \right)$$
2. Solve the equation $$\sin 2 \theta ( \operatorname { cosec } \theta - \sec \theta ) = 1$$ for \(0 < \theta < \frac { 1 } { 2 } \pi\). Give the answer correct to 3 significant figures.
3. Find \(\int \sin x \left( \operatorname { cosec } \frac { 1 } { 2 } x - \sec \frac { 1 } { 2 } x \right) \mathrm { d } x\).

8

1. Show that \(3 \sin 2 \theta \cot \theta \equiv 6 \cos ^ { 2 } \theta\).
2. Solve the equation \(3 \sin 2 \theta \cot \theta = 5\) for \(0 < \theta < \pi\).
3. Find the exact value of \(\int _ { \frac { 1 } { 4 } \pi } ^ { \frac { 1 } { 2 } \pi } 3 \sin x \cot \frac { 1 } { 2 } x \mathrm {~d} x\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

3

1. Show that \(( \sec x + \cos x ) ^ { 2 }\) can be expressed as \(\sec ^ { 2 } x + a + b \cos 2 x\), where \(a\) and \(b\) are constants to be determined.
2. Hence find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } ( \sec x + \cos x ) ^ { 2 } \mathrm {~d} x\).

6

1. Use the trapezium rule with three intervals to find an approximation to \(\int _ { 1 } ^ { 4 } \frac { 6 } { 1 + \sqrt { x } } \mathrm {~d} x\). Give your answer correct to 5 significant figures.
2. Find the exact value of \(\int _ { 1 } ^ { 4 } 2 \mathrm { e } ^ { \frac { 1 } { 2 } x - 2 } \mathrm {~d} x\).
3. \includegraphics[max width=\textwidth, alt={}, center]{2d6fc4c5-70ec-4cd8-9b48-59d5ce0e39b7-11_556_805_262_705} The diagram shows the curves \(y = \frac { 6 } { 1 + \sqrt { x } }\) and \(y = 2 \mathrm { e } ^ { \frac { 1 } { 2 } x - 2 }\) which meet at a point with \(x\)-coordinate 4. The shaded region is bounded by the two curves and the line \(x = 1\). Use your answers to parts (a) and (b) to find an approximation to the area of the shaded region. Give your answer correct to 3 significant figures.
4. State, with a reason, whether your answer to part (c) is an over-estimate or under-estimate of the exact area of the shaded region.

4

1. Find the exact value of \(\int _ { 0 } ^ { 2 } 6 \mathrm { e } ^ { 2 x + 1 } \mathrm {~d} x\).
2. Find \(\int \left( \tan ^ { 2 } x + 4 \sin ^ { 2 } 2 x \right) \mathrm { d } x\).

3 \includegraphics[max width=\textwidth, alt={}, center]{ed12a4fb-e3bf-4d00-ad09-9ba5be941dd5-04_531_739_258_703} The diagram shows the curve with equation \(y = 3 \sin x - 3 \sin 2 x\) for \(0 \leqslant x \leqslant \pi\). The curve meets the \(x\)-axis at the origin and at the points with \(x\)-coordinates \(a\) and \(\pi\).

1. Find the exact value of \(a\).
2. Find the area of the shaded region.

6 Show that \(\int _ { \frac { 1 } { 4 } \pi } ^ { \frac { 1 } { 3 } \pi } \left( 4 \cos ^ { 2 } 2 x + \frac { 1 } { \cos ^ { 2 } x } \right) \mathrm { d } x = \frac { 3 } { 4 } \sqrt { 3 } + \frac { 1 } { 6 } \pi - 1\).

3 \includegraphics[max width=\textwidth, alt={}, center]{4ce3208e-8ceb-4848-a9c7-fcda166319f4-04_458_892_269_614} The diagram shows part of the curve \(y = \frac { 6 } { 2 x + 3 }\). The shaded region is bounded by the curve and the lines \(x = 6\) and \(y = 2\). Find the exact area of the shaded region, giving your answer in the form \(a - \ln b\), where \(a\) and \(b\) are integers.

6

1. Show that \(4 \sin \left( \theta + \frac { 1 } { 3 } \pi \right) \cos \left( \theta - \frac { 1 } { 3 } \pi \right) \equiv \sqrt { 3 } + 2 \sin 2 \theta\).
2. Find the exact value of \(4 \sin \frac { 17 } { 24 } \pi \cos \frac { 1 } { 24 } \pi\).
3. Find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 8 } \pi } 4 \sin \left( 2 x + \frac { 1 } { 3 } \pi \right) \cos \left( 2 x - \frac { 1 } { 3 } \pi \right) \mathrm { d } x\).

3 \includegraphics[max width=\textwidth, alt={}, center]{a1ea242a-c7f4-46b0-b4b8-bd13b3880557-04_458_892_269_614} The diagram shows part of the curve \(y = \frac { 6 } { 2 x + 3 }\). The shaded region is bounded by the curve and the lines \(x = 6\) and \(y = 2\). Find the exact area of the shaded region, giving your answer in the form \(a - \ln b\), where \(a\) and \(b\) are integers.

3 \includegraphics[max width=\textwidth, alt={}, center]{76df3465-9617-4f2b-a8b7-f474b2817504-04_776_483_310_769} The diagram shows the curve with equation \(y = 8 \mathrm { e } ^ { - x } - \mathrm { e } ^ { 2 x }\). The curve crosses the \(y\)-axis at the point \(A\) and the \(x\)-axis at the point \(B\). The shaded region is bounded by the curve and the two axes.

1. Find the gradient of the curve at \(A\). \includegraphics[max width=\textwidth, alt={}, center]{76df3465-9617-4f2b-a8b7-f474b2817504-04_2715_35_141_2011}
2. Show that the \(x\)-coordinate of \(B\) is \(\ln 2\) and hence find the area of the shaded region.

7

1. Prove that \(2 \sin \theta \operatorname { cosec } 2 \theta \equiv \sec \theta\).
2. Solve the equation \(\tan ^ { 2 } \theta + 7 \sin \theta \operatorname { cosec } 2 \theta = 8\) for \(- \pi < \theta < \pi\). \includegraphics[max width=\textwidth, alt={}, center]{76df3465-9617-4f2b-a8b7-f474b2817504-12_2725_37_136_2010}
3. Find \(\int 8 \sin ^ { 2 } \frac { 1 } { 2 } x \operatorname { cosec } ^ { 2 } x \mathrm {~d} x\).
   If you use the following page to complete the answer to any question, the question number must be clearly shown. \includegraphics[max width=\textwidth, alt={}, center]{76df3465-9617-4f2b-a8b7-f474b2817504-14_2715_35_143_2012}

3 It is given that \(\int _ { a } ^ { 3 a } \frac { 2 } { 2 x - 5 } \mathrm {~d} x = \ln \frac { 7 } { 2 }\).
Find the value of the positive constant \(a\).

7 \includegraphics[max width=\textwidth, alt={}, center]{78a9b100-c3bd-4054-b539-ec8304440063-10_551_641_260_751} The diagram shows part of the curve with equation $$y = 4 \sin ^ { 2 } x + 8 \sin x + 3 ,$$ where \(x\) is measured in radians. The curve crosses the \(x\)-axis at the point \(A\) and the shaded region is bounded by the curve and the lines \(x = 0\) and \(y = 0\).

1. Find the exact \(x\)-coordinate of \(A\).
2. Find the exact gradient of the curve at \(A\).
3. Find the exact area of the shaded region.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

7

1. Express \(5 \sqrt { 3 } \cos x + 5 \sin x\) in the form \(R \cos ( x - \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { 1 } { 2 } \pi\).
2. As \(x\) varies, find the least possible value of $$4 + 5 \sqrt { 3 } \cos x + 5 \sin x$$ and determine the corresponding value of \(x\) where \(- \pi < x < \pi\).
3. Find \(\int \frac { 1 } { ( 5 \sqrt { 3 } \cos 3 \theta + 5 \sin 3 \theta ) ^ { 2 } } d \theta\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

5

1. Given that \(y = \tan ^ { 2 } x\), show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 2 \tan x + 2 \tan ^ { 3 } x\).
2. Find the exact value of \(\int _ { \frac { 1 } { 4 } \pi } ^ { \frac { 1 } { 3 } \pi } \left( \tan x + \tan ^ { 2 } x + \tan ^ { 3 } x \right) \mathrm { d } x\).

3 \includegraphics[max width=\textwidth, alt={}, center]{8beee722-7f86-454a-bc36-27e83f1483fd-04_684_455_260_845} The diagram shows the curve \(y = 2 + \mathrm { e } ^ { - 2 x }\). The curve crosses the \(y\)-axis at the point \(A\), and the point \(B\) on the curve has \(x\)-coordinate 1 . The shaded region is bounded by the curve and the line segment \(A B\). Find the exact area of the shaded region.

6

1. Find \(\int \left( \frac { 8 } { 4 x + 1 } + \frac { 8 } { \cos ^ { 2 } ( 4 x + 1 ) } \right) \mathrm { d } x\).
2. It is given that \(\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \left( 3 + 4 \cos ^ { 2 } \frac { 1 } { 2 } x + k \sin 2 x \right) \mathrm { d } x = 10\). Find the exact value of the constant \(k\).

3 \includegraphics[max width=\textwidth, alt={}, center]{b4a4082c-f3cd-47c5-8673-680dae9a22bd-04_684_455_260_845} The diagram shows the curve \(y = 2 + \mathrm { e } ^ { - 2 x }\). The curve crosses the \(y\)-axis at the point \(A\), and the point \(B\) on the curve has \(x\)-coordinate 1 . The shaded region is bounded by the curve and the line segment \(A B\). Find the exact area of the shaded region.