1. using the substitution u x

1. Using the substitution u = cos x + 1, or otherwise, show that

∫ +20

1coseπ

x sinx dx e(e – 1)

(Total 6 marks)

2. f(θ) = 4 cos2 θ – 3sin2 θ

(a) Show that f(θ) = .2cos27

21 θ+

(3)

(b) Hence, using calculus, find the exact value of ( )∫ 20

dfπ

θθθ

(7) (Total 10 marks)

3.

The diagram above shows a sketch of the curve with equation y = x 1n x, x ≥ 1. The finite region R, shown shaded in Figure 1, is bounded by the curve, the x-axis and the line x = 4.

The table shows corresponding values of x and y for y = x 1n x.

x 1 1.5 2 2.5 3 3.5 4

y 0 0.608 3.296 4.385 5.545

(a) Complete the table with the values of y corresponding to x = 2 and x = 2.5, giving your answers to 3 decimal places.

(2)

(b) Use the trapezium rule, with all the values of y in the completed table, to obtain an estimate for the area of R, giving your answer to 2 decimal places.

(4)

(c) (i) Use integration by parts to find xxx dn1∫ .

(ii) Hence find the exact area of R, giving your answer in the form ( ),2n141 ba +

where a and b are integers. (7)

(Total 13 marks

4. (a) Find .0,d69>

+∫ xx

xx

(2)

(b) Given that y = 8 at x = 1, solve the differential equation

x

yxxy 3

1

)69(dd +

=

giving your answer in the form y2 = g(x). (6)

(Total 8 marks)

5.

The diagram above shows a sketch of the curve C with parametric equations

x = 5t2 – 4, y = t(9 – t2)

The curve C cuts the x-axis at the points A and B.

(a) Find the x-coordinate at the point A and the x-coordinate at the point B. (3)

The region R, as shown shaded in the diagram above, is enclosed by the loop of the curve.

(b) Use integration to find the area of R. (6)

(Total 9 marks)

6. (a) Using the substitution x = 2 cos u, or otherwise, find the exact value of

( ) xxx

d2

1 –41

22∫

(7)

The diagram above shows a sketch of part of the curve with equation ,)4(

4142xx

y−

=

0 < x < 2.

The shaded region S, shown in the diagram above, is bounded by the curve, the x-axis and the lines with equations x = 1 and x = √2. The shaded region S is rotated through 2π radians about the x-axis to form a solid of revolution.

(b) Using your answer to part (a), find the exact volume of the solid of revolution formed. (3)

(Total 10 marks)

7.

The diagram above shows the finite region R bounded by the x-axis, the y-axis and the curve

with equation y = 3 cos .2

30,3

π≤≤

xx

The table shows corresponding values of x and y for y = 3 cos .3

x

x 0 8

3π 4

3π 8

9π 2

3π

y 3 2.77164 2.12132 0

(a) Complete the table above giving the missing value of y to 5 decimal places. (1)

(b) Using the trapezium rule, with all the values of y from the completed table, find an approximation for the area of R, giving your answer to 3 decimal places.

(4)

(c) Use integration to find the exact area of R. (3)

(Total 8 marks)

8. 3112)3)(1)(12(2–4)f(

++

++

+=

+++=

xC

xB

xA

xxxxx

(a) Find the values of the constants A, B and C. (4)

(b) (i) Hence find ∫ .d )f( xx (3)

(ii) Find ∫2

0d)f( xx in the form 1n k, where k is a constant.


9. (a) Find xx d)–5(∫

(2)

The diagram above shows a sketch of the curve with equation

y = (x – 1) √(5 – x), 1 ≤ x ≤ 5

(b) (i) Using integration by parts, or otherwise, find

xxx d)–5()1–(∫

(4)

(ii) Hence find xxx d)–5()1–(5

1∫ .

(2) (Total 8 marks)

10. (a) Using the identity cos2θ = 1 – 2sin2θ, find ∫ .dsin 2 θθ

(2)

The diagram above shows part of the curve C with parametric equations

x = tanθ, y = 2sin2θ, 0 ≤ θ < 2π

The finite shaded region S shown in the diagram is bounded by C, the line x = 3

1 and the

x-axis. This shaded region is rotated through 2π radians about the x-axis to form a solid of revolution.

(b) Show that the volume of the solid of revolution formed is given by the integral

∫ 60

2 dsinπ

θθk

where k is a constant. (5)

(c) Hence find the exact value for this volume, giving your answer in the form pπ2 + qπ √3, where p and q are constants.


11.

The diagram above shows part of the curve .x

y)4+(1

3= The region R is bounded by the

curve, the x-axis, and the lines x = 0 and x = 2, as shown shaded in the diagram above.

(a) Use integration to find the area of R. (4)

The region R is rotated 360° about the x-axis.

(b) Use integration to find the exact value of the volume of the solid formed. (5)

(Total 9 marks)

12. (a) Find .dtan 2 xx∫

(2)

(b) Use integration by parts to find ∫ 31x

ln x dx.

(4)

(c) Use the substitution u = 1 + ex to show that

,)1ln(21d

12

3

keeexe

e xxxx

x

+++−=+∫

where k is a constant. (7)

(Total 13 marks)

13. (a) Use integration by parts to find xx x de∫ .

(3)

(b) Hence find xx x de2∫ .

(3) (Total 6 marks)

14.

y

1

O a b x

The curve shown in the diagram above has equation )12(

1+

=x

y . The finite region bounded by

the curve, the x-axis and the lines x = a and x = b is shown shaded in the diagram. This region is rotated through 360° about the x-axis to generate a solid of revolution.

Find the volume of the solid generated. Express your answer as a single simplified fraction, in terms of a and b.

(Total 5 marks)

15. (i) Find ∫

xx d

21n .

(4)

(ii) Find the exact value of ∫ 2

4

2 dsinπ

π x .

(5) (Total 9 marks)

16. Use the substitution u = 2x to find the exact value of

( )∫+

1

0 2 .d12

2 xx

x

(Total 6 marks)

17. (a) Find ∫ .d2cos xxx .

(4)

(b) Hence, using the identity ∫−= xxxxx dcosdeduce,1cos22cos 22 .

(3) (Total 7 marks)

18. ( )( )( ) ( ) ( )12121212

142 2

−+

++≡

−++

xC

xBA

xxx

(a) Find the values of the constants A, B and C. (4)

(b) Hence show that the exact value of ( )( )( ) x

xxx

d1212

1422

1

2

∫ −++ is 2 + ln k, giving the value of

the constant k. (6)

(Total 10 marks)

19.

Figure 1

y

14

12

x0–

The curve with equation ( ) 21,

2131

−>+

= xx

y , is shown in Figure 1.

The region bounded by the lines 21,

41

=−= xx , the x-axis and the curve is shown shaded in

Figure 1.

This region is rotated through 360 degrees about the x-axis.

(a) Use calculus to find the exact value of the volume of the solid generated. (5)

Figure 2

B

A

Figure 2 shows a paperweight with axis of symmetry AB where AB = 3 cm. A is a point on the top surface of the paperweight, and B is a point on the base of the paperweight. The paperweight is geometrically similar to the solid in part (a).

(b) Find the volume of this paperweight. (2)

(Total 7 marks)

20. ( ) .de5

0

13 xI x∫ +=

(a) Given that y = e√(3x+1), complete the table with the values of y corresponding to x = 2, 3 and 4.

x 0 1 2 3 4 5

y e1 e2 e4 (2)

(b) Use the trapezium rule, with all the values of y in the completed table, to obtain an estimate for the original integral I, giving your answer to 4 significant figures.

(3)

(c) Use the substitution t = √(3x + 1) to show that I may be expressed as ∫b

a

t tkte d , giving

the values of a, b and k. (5)

(d) Use integration by parts to evaluate this integral, and hence find the value of I correct to 4 significant figures, showing all the steps in your working.


21.

y

O 2π x

The curve with equation, π20,2

sin3 ≤≤= xxy , is shown in the figure above. The finite

region enclosed by the curve and the x-axis is shaded.

(a) Find, by integration, the area of the shaded region. (3)

This region is rotated through 2π radians about the x-axis.

(b) Find the volume of the solid generated. (6)

(Total 9 marks)

22.

y

O x1

The figure above shows a sketch of the curve with equation y = (x – 1) ln x, x > 0.

(a) Complete the table with the values of y corresponding to x = 1.5 and x = 2.5.

x 1 1.5 2 2.5 3

y 0 ln 2 2ln 3 (1)

Given that ∫ −=3

1dln)1( xxxI

(b) use the trapezium rule

(i) with values of y at x = 1, 2 and 3 to find an approximate value for I to 4 significant figures,

(ii) with values of y at x = 1, 1.5, 2, 2.5 and 3 to find another approximate value for I to 4 significant figures.

(5)

(c) Explain, with reference to the figure above, why an increase in the number of values improves the accuracy of the approximation.

(1)

(d) Show, by integration, that the exact value of ∫ −3

1 23 3lnisdln)1( xxx .


23. f(x) = (x2 + 1) ln x, x > 0.

(a) Use differentiation to find the value of f'(x) at x = e, leaving your answer in terms of e. (4)

(b) Find the exact value of ∫e

1d)(f xx

(5) (Total 9 marks)

24.

.,4949)(f 2

32

2

±≠−+

= xxxx

(a) Find the values of the constants A, B and C such that

.,2323

)(f 23±≠

−+

++= x

xC

xBAx

(4)

(b) Hence find the exact value of

∫− −+1

1 2

2

d4949 x

xx

(5) (Total 9 marks)

25.

x

y

a

OB

12

a

A

The curve shown in the figure above has parametric equations

.6

0,sin,3cos π≤≤== ttaytax

The curve meets the axes at points A and B as shown.

The straight line shown is part of the tangent to the curve at the point A.

Find, in terms of a,

(a) an equation of the tangent at A, (6)

(b) an exact value for the area of the finite region between the curve, the tangent at A and the x-axis, shown shaded in the figure above.


26. Using the substitution u2 = 2x – 1, or otherwise, find the exact value of

∫ −√

5

1d

)12(3 xxx

(Total 8 marks)

27.

y x = ex

R

O x

y

1 3

The figure above shows the finite shaded region, R, which is bounded by the curve y = xex, the line x = 1, the line x = 3 and the x-axis.

The region R is rotated through 360 degrees about the x-axis.

Use integration by parts to find an exact value for the volume of the solid generated. (Total 8 marks)

28.

x

y

O

R

The curve shown in the figure above has parametric equations

x = t – 2 sin t, y = 1 – 2cos t, 0 ≤ t ≤ 2π

(a) Show that the curve crosses the x-axis where .3

5and3

ππ== tt

(2)

The finite region R is enclosed by the curve and the x-axis, as shown shaded in the figure above.

(b) Show that the area of R is given by the integral

( )∫ −35

3

.dcos21 2π

πtt

(3)

(c) Use this integral to find the exact value of the shaded area. (7)

(Total 12 marks)

29.

x

yC

1 3

(1, 2)

y =x

x + 1

O

Figure 1

Figure 1 shows part of the curve C with equation y = x

x 1+, x > 0.

The finite region enclosed by C, the lines x = 1, x = 3 and the x-axis is rotated through 360° about the x-axis to generate a solid S.

(a) Using integration, find the exact volume of S. (7)

x

yC

1 3

(1, 2)

y =x

x + 1

O

R

T

Figure 2

The tangent T to C at the point (1, 2) meets the x-axis at the point (3, 0). The shaded region R is bounded by C, the line x = 3 and T, as shown in Figure 2.

(b) Using your answer to part (a), find the exact volume generated by R when it is rotated through 360° about the x-axis.


30. (a) Use integration by parts to find

xxx d2cos⌡⌠ .

(4)

(b) Hence, or otherwise, find

xxx dcos2⌡⌠ .

(3) (Total 7 marks)

31. (a) Express )2)(32(

35+−

+xx

x in partial fractions.

(3)

(b) Hence find the exact value of xxx

x d)2)(32(

356

2⌡⌠

+−+

, giving your answer as a single

logarithm. (5)

(Total 8 marks)

32. Use the substitution x = sin θ to find the exact value of

xx

d)1(

121

23

02

⌡

⌠

−.

(Total 7 marks)

33.

x

y

R

0 0.2 0.4 0.6 0.8 1

The diagram shows the graph of the curve with equation

y = xe2x, x ≥ 0.

The finite region R bounded by the lines x = 1, the x-axis and the curve is shown shaded in the diagram.

(a) Use integration to find the exact value of the area for R. (5)

(b) Complete the table with the values of y corresponding to x = 0.4 and 0.8.

x 0 0.2 0.4 0.6 0.8 1

y = xe2x 0 0.29836 1.99207 7.38906

(1)

(c) Use the trapezium rule with all the values in the table to find an approximate value for this area, giving your answer to 4 significant figures.

(4)

34. (a) Use the formulae for sin (A ± B), with A = 3x and B = x, to show that 2 sin x cos 3x can be written as sin px – sin qx, where p and q are positive integers.

(3)

(b) Hence, or otherwise, find ∫ dxxx 3cossin2 . (2)

(c) Hence find the exact value of ∫ 65

2

3cossin2π

π dxxx

(2) (Total 7 marks)

35.

y

xO

C

P

Q

R

The diagram shows a sketch of part of the curve C with parametric equations

x = t2 + 1, y = 3(1 + t).

The normal to C at the point P(5, 9) cuts the x-axis at the point Q, as shown in the diagram.

(a) Find the x-coordinate of Q. (6)

(b) Find the area of the finite region R bounded by C, the line PQ and the x-axis. (9)

(Total 15 marks)

36. (a) Use integration by parts to show that

⌡⌠

+ xxx d

6cosec2 π

= – x cot

+

6πx + ln

+

6sin πx + c,

6π

− < x < 3π

.

(3)

(b) Solve the differential equation

sin2

+

6πx

xy

dd

= 2xy(y + 1)

to show that 1

ln21

+yy = –xcot

+

6πx + ln

+

6sin πx + c.

(6)

Given that y = 1 when x = 0,

(c) find the exact value of y when x = 12π

.


37.

y

O

R

x2 4

The diagram above shows part of the curve with equation

y = 4x – x6

, x > 0.

The shaded region R is bounded by the curve, the x-axis and the lines with equations x = 2 and x = 4. This region is rotated through 2π radians about the x-axis.

Find the exact value of the volume of the solid generated. (Total 8 marks)

38.

O x

y

R

1 4

C

The diagram above shows parts of the curve C with equation

y = x

x√+ 2 .

The shaded region R is bounded by C, the x-axis and the lines x = 1 and x = 4.

This region is rotated through 360° about the x-axis to form a solid S.

(a) Find, by integration, the exact volume of S. (7)

The solid S is used to model a wooden support with a circular base and a circular top.

(b) Show that the base and the top have the same radius. (1)

Given that the actual radius of the base is 6 cm,

(c) show that the volume of the wooden support is approximately 630 cm3. (2)

(Total 10 marks)

39. Use the substitution u = 1 + sin x and integration to show that

⌡⌠ sin x cos x (1 + sin x)5 dx =

421 (1 + sin x)6 [6 sin x – 1] + constant.

(Total 8 marks)

40.

O

y

a x

C

P

R

The diagram above shows a sketch of the curve C with parametric equations

x = 3t sin t, y = 2 sec t, 0 ≤ t < 2π

.

The point P(a, 4) lies on C.

(a) Find the exact value of a. (3)

The region R is enclosed by C, the axes and the line x = a as shown in the diagram above.

(b) Show that the area of R is given by

6 ⌡⌠ 3

0

π

(tan t + t) dt.

(4)

(c) Find the exact value of the area of R. (4)

(Total 11 marks)

41.

R

A B

C

x (metres)

y (metres)

The diagram above shows a cross-section R of a dam. The line AC is the vertical face of the dam, AB is the horizontal base and the curve BC is the profile. Taking x and y to be the horizontal and vertical axes, then A, B and C have coordinates (0, 0), (3π 2, 0) and (0, 30) respectively. The area of the cross-section is to be calculated.

Initially the profile BC is approximated by a straight line.

(a) Find an estimate for the area of the cross-section R using this approximation. (1)

The profile BC is actually described by the parametric equations.

x = 16t2 – π 2, y = 30 sin 2t, 4π

≤ t ≤ 2π

.

(b) Find the exact area of the cross-section R. (7)

(c) Calculate the percentage error in the estimate of the area of the cross-section R that you found in part (a).


42.

y

xO

C

A1

R

The diagram above shows the curve C with equation y = f(x), where

f(x) = x8

– x2, x > 0.

Given that C crosses the x-axis at the point A,

(a) find the coordinates of A. (3)

The finite region R, bounded by C, the x-axis and the line x = 1, is rotated through 2π radians about the x-axis.

(b) Use integration to find, in terms of π the volume of the solid generated. (7)

(Total 10 marks)

43.

x

y

O 1 2

R

The diagram above shows part of the curve with equation y = 1 + xc

, where c is a positive

constant.

The point P with x-coordinate p lies on the curve. Given that the gradient of the curve at P is −4,

(a) show that c = 4p2. (2)

Given also that the y-coordinate of P is 5,

(b) prove that c = 4. (2)

The region R is bounded by the curve, the x-axis and the lines x = 1 and x = 2, as shown in the diagram above. The region R is rotated through 360° about the x-axis.

(c) Show that the volume of the solid generated can be written in the form π(k + q ln 2), where k and q are constants to be found.


44.

y

x

M

O

P

N

R

The curve C with equation y = 2ex + 5 meets the y-axis at the point M, as shown in the diagram above.

(a) Find the equation of the normal to C at M in the form ax + by = c, where a, b and c are integers.

(4)

This normal to C at M crosses the x-axis at the point N(n, 0).

(b) Show that n = 14. (1)

The point P(ln 4, 13) lies on C. The finite region R is bounded by C, the axes and the line PN, as shown in the diagram above.

(c) Find the area of R, giving your answer in the form p + q ln 2, where p and q are integers to be found.


45.

A

O x

y

The diagram above shows a graph of y = x √ sin x, 0 < x < π. The maximum point on the curve is A.

(a) Show that the x-coordinate of the point A satisfies the equation 2 tan x + x = 0. (4)

The finite region enclosed by the curve and the x-axis is shaded as shown in the diagram above.

A solid body S is generated by rotating this region through 2π radians about the x-axis.

(b) Find the exact value of the volume of S. (7)

(Total 11 marks)

46. f(x) = )1()23(

252 xx −+

, x < 1.

(a) Express f(x) as a sum of partial fractions. (4)

(b) Hence find ∫ xx d)(f . (5)

(c) Find the series expansion of f(x) in ascending powers of x up to and including the term in x2. Give each coefficient as a simplified fraction.


47.

y

xO 41

R

The diagram above shows part of the curve with equation y = 1 + x2

1. The shaded region R,

bounded by the curve, the x-axis and the lines x = 1 and x = 4, is rotated through 360° about the x-axis. Using integration, show that the volume of the solid generated is π (5 + 2

1 ln 2). (Total 8 marks)

48.

y

O x

M

The diagram above shows the curve with equation y = 21

x e−2x.

(a) Find the x-coordinate of M, the maximum point of the curve. (5)

The finite region enclosed by the curve, the x-axis and the line x = 1 is rotated through 2π about the x-axis.

(b) Find, in terms of π and e, the volume of the solid generated. (7)

(Total 12 marks)

49. Find the volume generated when the region bounded by the curve with equation

y = 2 + x1

, the x-axis and the lines x = 21 and x = 4 is rotated through 360° about

the x-axis.

Give your answer in the form π(a + b ln 2), where a and b are rational constants.

…………………………..…….…………………………………………………………….. (Total 7 marks)

50. Given that y = 1 at x = π, solve the differential equation

xy

dd

= yx2 cos x, y > 0.

(Total 9 marks)

51.

g(x) = )2)(41(

85xx

x−+

+ .

(a) Express g(x) in the form )2()41( x

Bx

A−

++

, where A and B are constants to be found.

(3)

The finite region R is bounded by the curve with equation y = g(x), the coordinate axes and the line x = 2

1 .

(b) Find the area of R, giving your answer in the form a ln 2 + b ln 3. (7)

(Total 10 marks)

52. Use the substitution u2 = (x – 1) to find

⌡

⌠

−x

xx d

)1(

2,

giving your answer in terms of x. (Total 10 marks)

53. Use the substitution u = 4 + 3x2 to find the exact value of

xxx d

)34(22

022⌡

⌠+ .

(Total 6 marks)

54. f(x) = )21)(1(141

xxx

+−+

, 21

<x .

(a) Express f(x) in partial fractions. (3)

(b) Hence find the exact value of xx d)(f31

61

⌡⌠ , giving your answer in the form ln p, where

p is rational. (5)

(c) Use the binomial theorem to expand f(x) in ascending powers of x, up to and including the term in x3, simplifying each term.


55.

x

y

R

O 2

The diagram above shows the curve with equation

y = x2 sin ( 21 x), 0 < x ≤ 2π.

The finite region R bounded by the line x = π, the x-axis, and the curve is shown shaded in Fig 1.

(a) Find the exact value of the area of R, by integration. Give your answer in terms of π. (7)

The table shows corresponding values of x and y.

x π 4

5π

23π

4

7π

2π

y 9.8696 14.247 15.702 G 0

(b) Find the value of G. (1)

(c) Use the trapezium rule with values of x2 sin ( 21 x)

(i) at x = π, x = 2

3π and x = 2π to find an approximate value for the area R, giving

your answer to 4 significant figures,

(ii) at x = π, x = 4

5π, x =

23π

, x = 4

7π and x = 2π to find an improved

approximation for the area R, giving your answer to 4 significant figures. (5)

(Total 13 marks)

1. using the substitution u x

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