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

5

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

9 Let \(\mathrm { f } ( x ) = \frac { 3 x ^ { 2 } - 4 } { x ^ { 2 } ( 3 x + 2 ) }\).

1. Express \(\mathrm { f } ( x )\) in partial fractions.
2. Hence show that \(\int _ { 1 } ^ { 2 } \mathrm { f } ( x ) \mathrm { d } x = \ln \left( \frac { 25 } { 8 } \right) - 1\).

6

1. By first expanding \(\sin ( 2 x + x )\), show that \(\sin 3 x \equiv 3 \sin x - 4 \sin ^ { 3 } x\).
2. Hence, showing all necessary working, find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 3 } \pi } \sin ^ { 3 } x \mathrm {~d} x\).

3 Let \(f ( \theta ) = \frac { 1 - \cos 2 \theta + \sin 2 \theta } { 1 + \cos 2 \theta + \sin 2 \theta }\).

1. Show that \(\mathrm { f } ( \theta ) = \tan \theta\).
2. Hence show that \(\int _ { \frac { 1 } { 6 } \pi } ^ { \frac { 1 } { 4 } \pi } \mathrm { f } ( \theta ) \mathrm { d } \theta = \frac { 1 } { 2 } \ln \frac { 3 } { 2 }\).

9 Let \(\mathrm { f } ( x ) = \frac { 3 x ^ { 3 } + 6 x - 8 } { x \left( x ^ { 2 } + 2 \right) }\).

1. Express \(\mathrm { f } ( x )\) in the form \(A + \frac { B } { x } + \frac { C x + D } { x ^ { 2 } + 2 }\).
2. Show that \(\int _ { 1 } ^ { 2 } \mathrm { f } ( x ) \mathrm { d } x = 3 - \ln 4\).

6 \includegraphics[max width=\textwidth, alt={}, center]{79efa364-da5a-4888-85a9-dc4de1e0908e-3_543_825_287_660} The diagram shows the curve \(y = ( 3 - x ) \mathrm { e } ^ { - 2 x }\) and its minimum point \(M\). The curve intersects the \(x\)-axis at \(A\) and the \(y\)-axis at \(B\).

1. Calculate the \(x\)-coordinate of \(M\).
2. Find the area of the region bounded by \(O A , O B\) and the curve, giving your answer in terms of e.

10 \includegraphics[max width=\textwidth, alt={}, center]{9e3b4c96-0989-4ffb-bd74-0e73b79ca45a-3_430_807_1375_667} The diagram shows the curve \(y = x \cos 2 x\) for \(0 \leqslant x \leqslant \frac { 1 } { 4 } \pi\). The point \(M\) is a maximum point.

1. Show that the \(x\)-coordinate of \(M\) satisfies the equation \(1 = 2 x \tan 2 x\).
2. The equation in part (i) can be rearranged in the form \(x = \frac { 1 } { 2 } \tan ^ { - 1 } \left( \frac { 1 } { 2 x } \right)\). Use the iterative formula $$x _ { n + 1 } = \frac { 1 } { 2 } \tan ^ { - 1 } \left( \frac { 1 } { 2 x _ { n } } \right) ,$$ with initial value \(x _ { 1 } = 0.4\), to calculate the \(x\)-coordinate of \(M\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
3. Use integration by parts to find the exact area of the region enclosed between the curve and the \(x\)-axis from 0 to \(\frac { 1 } { 4 } \pi\).

5

1. Prove the identity \(\cos 4 \theta - 4 \cos 2 \theta + 3 \equiv 8 \sin ^ { 4 } \theta\).
2. Using this result find, in simplified form, the exact value of $$\int _ { \frac { 1 } { 6 } \pi } ^ { \frac { 1 } { 3 } \pi } \sin ^ { 4 } \theta \mathrm {~d} \theta$$

4 It is given that \(\mathrm { f } ( x ) = 4 \cos ^ { 2 } 3 x\).

1. Find the exact value of \(\mathrm { f } ^ { \prime } \left( \frac { 1 } { 9 } \pi \right)\).
2. Find \(\int \mathrm { f } ( x ) \mathrm { d } x\).

5

1. By differentiating \(\frac { 1 } { \cos x }\), show that if \(y = \sec x\) then \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \sec x \tan x\).
2. Show that \(\frac { 1 } { \sec x - \tan x } \equiv \sec x + \tan x\).
3. Deduce that \(\frac { 1 } { ( \sec x - \tan x ) ^ { 2 } } \equiv 2 \sec ^ { 2 } x - 1 + 2 \sec x \tan x\).
4. Hence show that \(\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \frac { 1 } { ( \sec x - \tan x ) ^ { 2 } } \mathrm {~d} x = \frac { 1 } { 4 } ( 8 \sqrt { } 2 - \pi )\).

5

1. Prove that \(\cot \theta + \tan \theta \equiv 2 \operatorname { cosec } 2 \theta\).
2. Hence show that \(\int _ { \frac { 1 } { 6 } \pi } ^ { \frac { 1 } { 3 } \pi } \operatorname { cosec } 2 \theta \mathrm {~d} \theta = \frac { 1 } { 2 } \ln 3\).

5

1. Prove the identity \(\tan 2 \theta - \tan \theta \equiv \tan \theta \sec 2 \theta\).
2. Hence show that \(\int _ { 0 } ^ { \frac { 1 } { 6 } \pi } \tan \theta \sec 2 \theta \mathrm {~d} \theta = \frac { 1 } { 2 } \ln \frac { 3 } { 2 }\).

9

1. By first expanding \(\cos ( 2 x + x )\), show that \(\cos 3 x \equiv 4 \cos ^ { 3 } x - 3 \cos x\).
2. Hence solve the equation \(\cos 3 x + 3 \cos x + 1 = 0\), for \(0 \leqslant x \leqslant \pi\).
3. Find the exact value of \(\int _ { \frac { 1 } { 6 } \pi } ^ { \frac { 1 } { 3 } \pi } \cos ^ { 3 } x \mathrm {~d} x\).

4

1. Find \(\int \tan ^ { 2 } 3 x \mathrm {~d} x\).
2. Find the exact value of \(\int _ { 0 } ^ { 1 } \frac { \mathrm { e } ^ { 3 x } + 4 } { \mathrm { e } ^ { x } } \mathrm {~d} x\). Show all necessary working.

4

1. Find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \left( 4 \sin 2 x + 2 \cos ^ { 2 } x \right) \mathrm { d } x\). Show all necessary working.
2. Use the trapezium rule with two intervals to find an approximation to \(\int _ { 2 } ^ { 8 } \sqrt { } ( \ln ( 1 + x ) ) \mathrm { d } x\)

5

1. Find the quotient and remainder when \(2 x ^ { 3 } + x ^ { 2 } - 8 x\) is divided by ( \(2 x + 1\) ).
2. Hence find the exact value of \(\int _ { 0 } ^ { 3 } \frac { 2 x ^ { 3 } + x ^ { 2 } - 8 x } { 2 x + 1 } \mathrm {~d} x\), giving the answer in the form \(\ln \left( k \mathrm { e } ^ { a } \right)\) where \(k\) and \(a\) are constants.

5 Given that \(\int _ { 0 } ^ { a } 6 \mathrm { e } ^ { 2 x + 1 } \mathrm {~d} x = 65\), find the value of \(a\) correct to 3 decimal places.

8

1. Show that \(\sin 2 x \cot x \equiv 2 \cos ^ { 2 } x\).
2. Using the identity in part (i),
   1. find the least possible value of $$3 \sin 2 x \cot x + 5 \cos 2 x + 8$$ as \(x\) varies,
   2. find the exact value of \(\int _ { \frac { 1 } { 8 } \pi } ^ { \frac { 1 } { 6 } \pi } \operatorname { cosec } 4 x \tan 2 x \mathrm {~d} x\).

6

1. Show that \(\int _ { 1 } ^ { 4 } \left( \frac { 2 } { x } + \frac { 2 } { 2 x + 1 } \right) \mathrm { d } x = \ln 48\).
2. Find \(\int \sin 2 x ( \cot x + 2 \operatorname { cosec } x ) \mathrm { d } x\).

6

1. Find the value of \(\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } ( \sin 2 x + \cos x ) \mathrm { d } x\).
2. \includegraphics[max width=\textwidth, alt={}, center]{9894d97f-3b7b-4dbe-b94a-2c8415442038-3_517_880_422_669} The diagram shows part of the curve \(y = \frac { 1 } { x + 1 }\). The shaded region \(R\) is bounded by the curve and by the lines \(x = 1 , y = 0\) and \(x = p\).
   1. Find, in terms of \(p\), the area of \(R\).
   2. Hence find, correct to 1 decimal place, the value of \(p\) for which the area of \(R\) is equal to 2 .

7 \includegraphics[max width=\textwidth, alt={}, center]{25dffd43-9456-449b-be77-8402109ee603-3_608_672_283_733} The diagram shows the curve \(y = 2 \mathrm { e } ^ { x } + 3 \mathrm { e } ^ { - 2 x }\). The curve cuts the \(y\)-axis at \(A\).

1. Write down the coordinates of \(A\).
2. Find the equation of the tangent to the curve at \(A\), and state the coordinates of the point where this tangent meets the \(x\)-axis.
3. Calculate the area of the region bounded by the curve and by the lines \(x = 0 , y = 0\) and \(x = 1\), giving your answer correct to 2 significant figures.

8

1. Express \(\cos \theta + \sin \theta\) in the form \(R \cos ( \theta - \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { 1 } { 2 } \pi\), giving the exact values of \(R\) and \(\alpha\).
2. Hence show that $$\frac { 1 } { ( \cos \theta + \sin \theta ) ^ { 2 } } = \frac { 1 } { 2 } \sec ^ { 2 } \left( \theta - \frac { 1 } { 4 } \pi \right)$$
3. By differentiating \(\frac { \sin x } { \cos x }\), show that if \(y = \tan x\) then \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \sec ^ { 2 } x\).
4. Using the results of parts (ii) and (iii), show that $$\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \frac { 1 } { ( \cos \theta + \sin \theta ) ^ { 2 } } \mathrm {~d} \theta = 1$$

6 A curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \mathrm { e } ^ { 2 x } - 2 \mathrm { e } ^ { - x }\). The point \(( 0,1 )\) lies on the curve.

1. Find the equation of the curve.
2. The curve has one stationary point. Find the \(x\)-coordinate of this point and determine whether it is a maximum or a minimum point.

7 \includegraphics[max width=\textwidth, alt={}, center]{d527d21f-0ab5-40fa-8cfd-ebfb4aba0a87-3_493_863_264_641} The diagram shows the part of the curve \(y = \sin ^ { 2 } x\) for \(0 \leqslant x \leqslant \pi\).

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \sin 2 x\).
2. Hence find the \(x\)-coordinates of the points on the curve at which the gradient of the curve is 0.5 . [3]
3. By expressing \(\sin ^ { 2 } x\) in terms of \(\cos 2 x\), find the area of the region bounded by the curve and the \(x\)-axis between 0 and \(\pi\).

6 \includegraphics[max width=\textwidth, alt={}, center]{4029c46c-50a1-4d23-bc29-589417a6b7f5-3_501_497_269_826} The diagram shows the part of the curve \(y = \frac { \mathrm { e } ^ { 2 x } } { x }\) for \(x > 0\), and its minimum point \(M\).

1. Find the coordinates of \(M\).
2. Use the trapezium rule with 2 intervals to estimate the value of $$\int _ { 1 } ^ { 2 } \frac { \mathrm { e } ^ { 2 x } } { x } \mathrm {~d} x$$ giving your answer correct to 1 decimal place.
3. State, with a reason, whether the trapezium rule gives an under-estimate or an over-estimate of the true value of the integral in part (ii).
4. Given that \(y = \tan 2 x\), find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
5. Hence, or otherwise, show that $$\int _ { 0 } ^ { \frac { 1 } { 6 } \pi } \sec ^ { 2 } 2 x \mathrm {~d} x = \frac { 1 } { 2 } \sqrt { } 3$$ and, by using an appropriate trigonometrical identity, find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 6 } \pi } \tan ^ { 2 } 2 x \mathrm {~d} x\).
6. Use the identity \(\cos 4 x \equiv 2 \cos ^ { 2 } 2 x - 1\) to find the exact value of $$\int _ { 0 } ^ { \frac { 1 } { 6 } \pi } \frac { 1 } { 1 + \cos 4 x } \mathrm {~d} x$$