1.08d Evaluate definite integrals: between limits

5

1. Express \(\cos \theta + ( \sqrt { } 3 ) \sin \theta\) 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 \(\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \frac { 1 } { ( \cos \theta + ( \sqrt { } 3 ) \sin \theta ) ^ { 2 } } \mathrm {~d} \theta = \frac { 1 } { \sqrt { } 3 }\).

4

1. Using the expansions of \(\cos ( 3 x - x )\) and \(\cos ( 3 x + x )\), prove that $$\frac { 1 } { 2 } ( \cos 2 x - \cos 4 x ) \equiv \sin 3 x \sin x$$
2. Hence show that $$\int _ { \frac { 1 } { 6 } \pi } ^ { \frac { 1 } { 3 } \pi } \sin 3 x \sin x \mathrm {~d} x = \frac { 1 } { 8 } \sqrt { } 3$$

9

1. Prove the identity \(\cos 4 \theta + 4 \cos 2 \theta \equiv 8 \cos ^ { 4 } \theta - 3\).
2. Hence
   1. solve the equation \(\cos 4 \theta + 4 \cos 2 \theta = 1\) for \(- \frac { 1 } { 2 } \pi \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\),
   2. find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \cos ^ { 4 } \theta \mathrm {~d} \theta\).

9 By first expressing \(\frac { 4 x ^ { 2 } + 5 x + 3 } { 2 x ^ { 2 } + 5 x + 2 }\) in partial fractions, show that $$\int _ { 0 } ^ { 4 } \frac { 4 x ^ { 2 } + 5 x + 3 } { 2 x ^ { 2 } + 5 x + 2 } \mathrm {~d} x = 8 - \ln 9$$

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

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

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.

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 \(\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \frac { 1 } { ( \sec x - \tan x ) ^ { 2 } } \mathrm {~d} x = \frac { 1 } { 4 } ( 8 \sqrt { } 2 - \pi )\).

7 \includegraphics[max width=\textwidth, alt={}, center]{adbef77f-e2ac-40ce-a56b-cf6776534ec1-3_543_1091_1402_529} The diagram shows part of the curve \(y = \sin ^ { 3 } 2 x \cos ^ { 3 } 2 x\). The shaded region shown is bounded by the curve and the \(x\)-axis and its exact area is denoted by \(A\).

1. Use the substitution \(u = \sin 2 x\) in a suitable integral to find the value of \(A\).
2. Given that \(\int _ { 0 } ^ { k \pi } \left| \sin ^ { 3 } 2 x \cos ^ { 3 } 2 x \right| \mathrm { d } x = 40 A\), find the value of the constant \(k\).

5 It is given that \(a\) is a positive constant such that $$\int _ { 0 } ^ { a } \left( 1 + 2 x + 3 \mathrm { e } ^ { 3 x } \right) \mathrm { d } x = 250$$

1. Show that \(a = \frac { 1 } { 3 } \ln \left( 251 - a - a ^ { 2 } \right)\).
2. Use an iterative formula based on the equation in part (i) to find the value of \(a\) correct to 4 significant figures. Give the result of each iteration to 6 significant figures.

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.

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

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

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

8

1. By first expanding \(\cos ( 2 x + x )\), show that $$\cos 3 x \equiv 4 \cos ^ { 3 } x - 3 \cos x$$
2. Hence show that $$\int _ { 0 } ^ { \frac { 1 } { 6 } \pi } \left( 2 \cos ^ { 3 } x - \cos x \right) d x = \frac { 5 } { 12 }$$

4 Find the exact value of the positive constant \(k\) for which $$\int _ { 0 } ^ { k } e ^ { 4 x } d x = \int _ { 0 } ^ { 2 k } e ^ { x } d x$$

4

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

7 \includegraphics[max width=\textwidth, alt={}, center]{0900b607-6136-4bf7-a42e-6824d1a21e43-3_451_451_255_845} The diagram shows part of the curve \(y = 8 x + \frac { 1 } { 2 } \mathrm { e } ^ { x }\). The shaded region \(R\) is bounded by the curve and by the lines \(x = 0 , y = 0\) and \(x = a\), where \(a\) is positive. The area of \(R\) is equal to \(\frac { 1 } { 2 }\).

1. Find an equation satisfied by \(a\), and show that the equation can be written in the form $$a = \sqrt { } \left( \frac { 2 - \mathrm { e } ^ { a } } { 8 } \right)$$
2. Verify by calculation that the equation \(a = \sqrt { } \left( \frac { 2 - \mathrm { e } ^ { a } } { 8 } \right)\) has a root between 0.2 and 0.3.
3. Use the iterative formula \(a _ { n + 1 } = \sqrt { } \left( \frac { 2 - \mathrm { e } ^ { a _ { n } } } { 8 } \right)\) to determine this root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

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.

6 \includegraphics[max width=\textwidth, alt={}, center]{6bf7ba66-8362-4ac0-8e5c-3f88a3ccdf86-10_351_488_264_826} The diagram shows the curve with equation \(y = \sqrt { } \left( 1 + 3 \cos ^ { 2 } \left( \frac { 1 } { 2 } x \right) \right)\) for \(0 \leqslant x \leqslant \pi\). The region \(R\) is bounded by the curve, the axes and the line \(x = \pi\).

1. Use the trapezium rule with two intervals to find an approximation to the area of \(R\), giving your answer correct to 3 significant figures.
2. The region \(R\) is rotated completely about the \(x\)-axis. Without using a calculator, find the exact volume of the solid produced.

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

7

1. Show that \(\cos ^ { 4 } \theta - \sin ^ { 4 } \theta \equiv \cos 2 \theta\).
2. Hence find the exact value of \(\int _ { - \frac { 1 } { 8 } \pi } ^ { \frac { 1 } { 8 } \pi } \left( \cos ^ { 4 } \theta - \sin ^ { 4 } \theta + 4 \sin ^ { 2 } \theta \cos ^ { 2 } \theta \right) \mathrm { d } \theta\).